Harmonic Resonance Mode Analysis and Application for Offshore Wind Power Plants
Henrik Andreas Brantsæter
Master of Energy and Environmental Engineering Submission date: December 2015 Supervisor: Elisabetta Tedeschi, ELKRAFT
Norwegian University of Science and Technology Department of Electric Power Engineering
Preface
This thesis is the result of my 11th and final semester as a master student at the De- partment of Electric Power Engineering, NTNU.
Lukasz Kocewiak at DONG Energy Wind Power has co-supervised my work for this thesis. He initiated the cooperation which led to a specialisation project and master thesis related to harmonics in offshore wind power plants. I would like to thank him for sharing his exceptional knowledge and experience in conference calls, emails, and on the occasions we have met in Trondheim.
Secondly, I would like to thank Atle Rygg Ardal˚ for guiding me in choosing a topic for my specialisation project, and for providing invaluable contributions and support to a paper which was submitted in relation to the specialisation project.
The main supervisor for this master thesis has been Prof. Elisabetta Tedeschi. I would like to express my sincere gratitude to her for taking an interest in my work, finding time to give valuable feedback, and for being enthusiastic and supportive.
Lastly, I would like to thank my family for inspiration, love and support.
NTNU Trondheim, December 2015 Henrik Andreas Broch Brantsæter
i
Abstract
This thesis is centred around a form of resonance analysis for power systems known as Harmonic Resonance Mode Analysis (HRMA). It is a form of eigenvalue analysis and has some parallels to the modal analysis commonly associated with small signal stability. Modes in small signal stability analysis characterise the transient response of the system to a small disturbance, and are identified from the state matrix of the linearised state space description. Modes in HRMA characterise the parallel resonances of a power system, and are identified from the admittance matrix. Participation factors, which are derived from eigenvectors of the admittance matrix at resonance frequencies, indicate the observability of modes in the impedance measured at individual buses. In addition, sensitivity indices quantify the change in amplitude and frequency of modes when parameters are adjusted in the admittance matrix.
A part of this thesis is dedicated to replicating HRMA results found in published papers. By validating published results for two different test cases, the computer implementation for carrying out the analysis is verified. Obtained results are generally consistent with those in published papers. Some critical considerations in implementation of HRMA are identified and discussed, perhaps the most important of which is the issue of modal switching.
HRMA is also used to investigate resonances in a system modelled on the real offshore wind power plant of Anholt in Denmark. HRMA is based on the nodal admittance matrix, and power system components are modelled as equivalent admittances. Non- linearities and frequency dependencies can be included more easily than in a state space representation. Examining the impact of non-linear components in the admittance ma- trix on HRMA results is a central contribution from this thesis. The sensitivity of results with respect to degree of aggregation of the wind power plant model is also investigated.
Modern wind turbines are connected to power systems via full-scale frequency convert- ers. Such converters can be modelled for harmonic analyses as equivalent impedances which are determined primarily from the converter current control loops. An equivalent impedance representing the grid-side converter of a wind turbine generator as seen from the grid is derived, and later such impedances are incorporated in the HRMA. iii iv
The resonance modes in the investigated large offshore wind power plant are sensitive to modelling of wind turbine main circuit components such as shunt harmonic filters, step- up transformers and main reactors. Suitable modelling of these electrical components is therefore important for obtaining reliable result from HRMA. Moreover, the resonance modes are highly dependent on the number of turbines in operation. Sammendrag
Denne oppgaven omhandler en metode for analyse av resonanser i kraftsystemer kjent som HRMA (eng. Harmonic Resonance Mode Analysis). Innen kraftsystemanalyse er det velkjent at egenverdiene til den lineariserte tilstandsrombeskrivelsen, ogs˚akjent som systemets modi, bestemmer systemets transiente respons til sm˚aforstyrrelser. HRMA har visse paralleller til denne formen for analyse. Det har vist seg at egenverdiene til admittansmatrisen blir svært sm˚aved frekvenser som er knyttet til parallelle resonanser. Mininum i egenverdienes frekvensresponser kan derfor defineres som systemets resonans- modi (eng. resonance modes). Impedansen m˚alt fra de ulike nodene i kretsen kan knyttes til de ulike resonans-modi gjennom admittansmatrisens egenvektorer. I tillegg er det definert indekser som indikerer hvordan frekvensen og amplituden til resonans-modiene endres ved ved ˚ajustere parametere i admittansmatrisen.
Ett kapittel i denne oppgaven er avsatt til ˚agjenskape HRMA-resultater for to ulike testsystemer som av andre har blitt presentert i publiserte artikler. P˚aden m˚aten verifiseres implementering av metoden i MATLAB. Resultater som presenteres fra de to testsystemene i denne oppgaven er i overensstemmelse med de resultatene som finnes i tidligere publiserte artikler. Noen viktige betraktninger rundt implementeringen av HRMA som ikke finnes i de nevnte artikler blir identifisert og diskutert.
HRMA benyttes videre til ˚aanalysere parallelle resonanser i ett system modellert etter den reelle hav-vindparken ved øyen Anholt i Danmark. HRMA baserer seg p˚aadmittans- matrisen til systemet som analyseres, og hver komponent modelleres som en ekvivalent admittans. Det er enklere ˚ainkludere ulineære komponenter i en slik admittansmodell enn i en tilstandsrombeskrivelse. Undersøkelser av hvordan HRMA-resultatene p˚avirkes n˚arulineære komponenter inkluderes i admittansbeskrivelsen er ett sentralt bidrag fra denne oppgaven. Det undersøkes ogs˚ahvordan resultatene p˚avirkes av ulike grader av aggregering av vindparkmodellen.
Moderne vindturbiner knyttes til nettet via fullskala frekvensomformere. For harmonisk analyse fremst˚arslike omformere som frekvensavhengige impedanser som særlig p˚avirkes av omformerens strømregulator. Fra ett teoretisk perspektiv vises det i denne oppgaven
v vi hvordan en slik ekvivalent impedans kan utledes, og videre inkluderes disse impedansene i HRMA av vindparkmodellen.
Analyse av vindparken viser at resonans-modiene er sensitive med hensyn til modellering av komponentene som finnes i nær tilknytning til hver enkelt turbin, slik som harmoniske filtre og transformatorer. Det er derfor viktig ˚amodellere slike komponenter med omhu for harmonisk analyse. Videre er vindparkens modi svært avhengig av antall turbiner som til en hver tid produserer. Contents
Preface i
Abstract iii
Sammendragv
Contents vii
List of Figures ix
List of Tables xi
Abbreviations xii
Symbols xiii
1 Introduction1 1.1 Problem Background and Motivation...... 1 1.2 Limitations of Scope...... 2 1.3 Relation to Specialisation Project...... 2 1.4 Structure...... 3
2 Harmonic Analysis Theory4 2.1 Harmonics...... 4 2.2 Harmonic Resonance...... 6 2.3 Methods of Harmonic Analysis...... 7 2.4 Harmonic Resonance Mode Analysis (HRMA)...... 12 2.5 Comparing Modal Analysis Techniques...... 22
3 Modelling for Harmonic Resonance Studies 24 3.1 Overhead Lines and Cables...... 24 3.2 Transformers...... 26 3.3 Aggregate Loads...... 27 3.4 VSC Grid-side Harmonic Emission Filters...... 28 3.5 Harmonic Sources/Full-Scale VSC Wind Turbines...... 29 3.6 Wind Power Plant Model Aggregation...... 33
vii Contents viii
4 Test Cases and HRMA Implementation 35 4.1 Three Bus Case...... 35 4.2 IEEE 14 Bus Case...... 41 4.3 HRMA Implementation...... 47
5 Offshore WPP Resonance Analysis 53 5.1 Anholt Offshore WPP Layout...... 53 5.2 9 Bus Model HRMA...... 59 5.3 9 Bus Model with WTG Terminal Filters...... 69
6 Discussion 75 6.1 General Remarks...... 75 6.2 Wind Power Plant Analysis...... 77
7 Conclusions 79 7.1 Further Work...... 79
A HRMA Test Case Input Data and Results 80 A.1 Input Data IEEE 14 Bus Test System...... 80 A.2 Test Case Results...... 82
B Details of HRMA 83 B.1 2-norm of Eigenvectors...... 83 B.2 Modal Sensitivity Indices Differential Expressions...... 83
C Offshore WPP Analysis 85 C.1 Anholt WPP Collector Array Layout...... 85 C.2 Input Data...... 86 C.3 Results 9 Bus Model with Nominal Cable Models...... 86
D LCL Filter Design Grid-tied VSC 87
Bibliography 92 List of Figures
2.1 Parallel resonant circuit...... 6 2.2 Series resonant circuit...... 7 2.3 Harmonic analysis methods...... 8 2.4 Illustration of driving point and transfer impedance...... 10 2.5 Modal resonance peak movement following circuit parameter change... 17
3.1 π-equivalent line/cable model...... 25 3.2 Transformer T-model...... 26 3.3 Aggregate load models for harmonic studies...... 27 3.4 LCL emission filter for grid-tied VSC...... 28 3.5 Grid-tied VSC with indication of current control and grid impedance... 29 3.6 VSC current control loops with current feed-back and voltage feed-forward 30 3.7 VSC current control loop with background voltage disturbance...... 31 3.8 Norton equivalent harmonic model...... 32
4.1 Three bus test case layout and data...... 35 4.2 Three bus example frequency scan...... 36 4.3 Three bus example modal impedance scan...... 37 4.4 Three bus test case, frequency sensitivity of h=51.61 pu resonance mode. 39 4.5 Three bus test case, frequency sensitivity of h=9.62 pu resonance mode.. 40 4.6 Movement of 9.62 pu resonance mode peak due to parameter change... 40 4.7 Single line diagram of IEEE 14 bus system...... 42 4.8 Nodal impedance scan IEEE 14 bus system...... 43 4.9 Modal impedance scan of IEEE 14 bus system...... 43 4.10 Sensitivity indices for h=4.212 resonance...... 44 4.11 Movement of h=4.212 pu modal resonance with parameter variation... 45 4.12 Frequency scan cable model comparison...... 46 4.13 Modal impedance scan of IEEE 14 bus system using nominal π-equivlaent line models...... 47 4.14 Modal switching illustration...... 48 4.15 Modal impedance envelope with real matrix power algorithm...... 51
5.1 Anholt WPP single line diagram...... 54 5.2 Anholt 33 bus model single line diagram...... 56 5.3 Anholt 15 bus single line diagram...... 57 5.4 Anholt 9 bus single line diagram...... 57 5.5 Modal envelope of 9, 15 and 33 bus models...... 58 5.6 Modal impedance scan of 9 bus WPP model w. nominal cable models.. 59
ix List of Figures x
5.7 Sensitivivity indices of most involved parameters in 16.84 pu resoance.. 61 5.8 Sensitivivity indices of most involved parameters in 16.25 pu resoance.. 61 5.9 Sensitivivity indices of most involved parameters in 15.62 pu resoance.. 61 5.10 Driving point impedance around 4.36 pu resonance...... 62 5.11 Sensitivivity index of most involved parameters in 4.36 pu resoance.... 62 5.12 Driving point impedance obtained at end of collector radials 9 bus WPP model...... 63 5.13 Impedance and modal scan with radial symmetry...... 64 5.14 Modal impedance scan of Anholt 9 bus system...... 65 5.15 Driving point impedance at buses 7, 8 and 9 near resonance frequecies due to offshore transformers and collector cables...... 66 5.16 Sensitivity indices of 15.86 pu and 16.57 pu Anholt 9 bus model...... 67 5.17 9 bus Anholt modal envelope with skin effect...... 68 5.18 WTG main circuit...... 69 5.19 Modal envelope with WTG main circuit included...... 70 5.20 Modal impedance scan 9 WPP model...... 70 5.21 Modal impedance envelope as function of number of operating turbines. 72 5.22 Modal impedance scan 33 bus WPP model...... 73 5.23 Modal impedance scan 9 WPP model w. converter control equivalent impedance...... 74
6.1 Root locus style presentation of resonance modes...... 78
C.1 Anholt offshore WPP collector array layout...... 85
D.1 LCL filter for grid-tied VSC with nomenclature...... 87 D.2 Ripple attenuation as function of capacitance C and inductance L1, two- level VSC...... 89 D.3 Ripple attenuation as function of capacitance C and inductance L1, three- level VSC...... 90 List of Tables
4.1 Three bus case driving point impedance at resonance frequencies..... 37 4.2 Three bus case participation factors...... 38 4.3 IEEE 14 bus system modal resonance key results...... 44
5.1 Anholt 9 bus WPP participation factors with nominal cable models... 59 5.3 Anholt 9 bus WPP participation factors with long-line cable models... 65 5.4 Anholt 9 bus WPP participation factors with skin effect...... 68 5.5 Anholt 9 bus WPP participation factors with LCL filters...... 71
A.1 IEEE 14 bus test case stedy state load flow...... 80 A.2 IEEE 14 bus test case generator input data...... 80 A.3 IEEE 14 bus test case branch parameters...... 81 A.4 IEEE 14 bus system, participation factors for all buses and resonances... 82
C.1 Submarine cable data for three-core submarine array cables of nominal voltage 33 kV...... 86
D.1 Per unit base values wind turbine emission filter design...... 87
xi Abbreviations
AC Alternating Current CIGRE Conseil International des Grands R´eseaux E´lectriques (The International Council on Large Electric Systems) DC Direct Current DSO Distribution System Operator IEEE Institute of Electrical and Electronics Engineers HRMA Harmonic Resonance Mode Analysis HVAC High Voltage Alternating Current HVDC High Voltage Direct Current MATLAB Matrix Laboratory (computing environment) PCC Point of Common Coupling PF Participation Factor PWM Pulse Width Modulation SVC Static VAR (reactive power) Compensator TSO Transmission System Operator VSC Voltage Source Converter WPF Weighted Participation Factor WPP Wind Power Plant WTG Wind Turbine Generator
xii Symbols
f frequency pu or Hz
f1 fundamental frequency pu or Hz h harmonic number pu ω angular frequency rad s−1 R Resistance pu or Ω L Inductance pu or H C Capacitance pu or F X Reactance pu or Ω G Conductance pu or S B Susceptance pu or S
[Y ], [Yf ] Nodal admittance matrix (at frequency f) [Z], [Y ]−1 Nodal impedance matrix
[V ], [Vf ] Nodal voltage vector (at frequency f)
[I], [If ] Nodal current vector (at frequency f) [U] Modal voltage vector [J] Modal current vector
Vi Nodal voltage at bus i pu or V
Ii Nodal current at bus i pu or A
Zii Driving point impedance bus i pu or Ω
Yii Driving point admittance bus i pu or S
Zm,k Modal impedance number k pu or Ω
λk Eigenvalue number k
lk Left eigenvector of k-th eigenvalue
tk Right eigenvector of k-th eigenvalue [Λ] Diagonal eigenvalue matrix
xiii Symbols xiv
[L] left eigenvector matrix [T ] right eigenvector matrix α Parameter of some power system component Al Aluminium Cu Copper 1 Introduction
In the context of worldwide commitments on curbing greenhouse gas emissions and transitioning to low carbon economies, governing bodies such as the EU, and inter- governmental agencies such as IEA (International Energy Agency), envision electricity generation in future power systems will be predominantly from renewable sources [1]. Offshore wind represents one of the most stable sources of renewable energy in many parts of the world. In comparison to onshore wind, there are benefits like reduced ”Not In My Back Yard”-opposition, stronger and more consistent winds, and fewer constraints with respect to structure size and noise. Though harnessing the offshore wind energy is technically challenging and more expensive compared to onshore wind in today’s mar- ket, offshore wind technology is rapidly emerging and arguably has important long-term potential. Realising the full potential of offshore wind will require support from pol- icy makers in facilitating continued cost reduction through research and development of turbine and transmission technology, supply chain optimisation, and operation & maintenance [2].
1.1 Problem Background and Motivation
Power system voltage and current waveforms are ideally sinusoidal, continuous and of constant frequency. Power quality can be defined as the combination of voltage quality and current quality, and is concerned with the deviation of voltage and current waveforms from the ideal. Harmonics are one of the major types of disturbances to the ideal waveforms covered by the broader term of power quality disturbances [3].
State of the art offshore Wind Power Plants (WPP) are large and complex structures including Wind Turbine Generators (WTG), array cable systems and HVAC or HVDC export systems. Such WPPs represent new challenges with respect to prediction and mitigation of harmonic emission and propagation [4]. Harmonic resonances in power systems have the potential to amplify harmonic distortion. In offshore WPPs this is- sue is exacerbated by the heavily armoured submarine cables required for collection
1 Chapter 1. Introduction 2 and transmission of power. The capacitance of such cables can introduce problematic resonances at relatively low frequencies.
Although the cause of harmonic resonance is well understood, the tools and techniques available for analysis of such phenomena are arguably quite limited. Harmonic Reso- nance Mode Analysis (HRMA) is a relatively new application of eigenvalue analysis, and is also known as Resonance Mode Analysis (RMA) or simply modal analysis. It has been suggested that this method can be used in conjunction with traditional harmonic analysis to gain better understanding of harmonic resonances and solve harmonic prob- lems more effectively [5–8]. However, the benefits and limitations of this method are not entirely clear from the handful of published papers which are related to it.
1.2 Limitations of Scope
The objective of this thesis is to investigate the potential of HRMA as a tool for harmonic resonance analysis. It aims to provide a comprehensive review of the HRMA concept, including verification of previous works on the topic. Application of the HRMA method to a large offshore wind power plant facilitates independent contributions. Examination of the sensitivity of HRMA results with respect to non-linearities and degree of model simplification constitutes the main independent contribution of the thesis.
The HRMA method is usually associated to the nodal admittance matrix, also known as Y-bus matrix, of the system under study, and only this kind of system description is considered in the thesis. As a consequence, HRMA is limited to the study of parallel resonances. An alternative system description, known as the loop impedance matrix, in principle allows for equivalent investigation of series resonances. HRMA based on loop impedance description has however received little attention in published papers, and is considered outside the scope of this thesis.
Quantifying actual waveform distortion due to harmonic emission and propagation is considered outside the scope of the thesis.
1.3 Relation to Specialisation Project
The author wrote a specialisation project in the autumn of 2014, titled ”Wind Turbine and Offshore Wind Power Plant Modelling for System Level Harmonic Studies” [9]. Some findings from the specialisation project were presented in a paper [10]. The project’s objective was to create a foundation for system level time-domain harmonic studies of offshore WPPs. It consisted of the following main components: Chapter 1. Introduction 3
• A literature study covering topics such as offshore WPP layout, power electron- ics for variable-speed wind turbines, harmonic waveforms, harmonic resonance, harmonic indices, passive filters, standards and grid codes related to harmonic distortion.
• Design of emission filter (LCL filter) for the grid connection of wind turbines with full-scale converters.
• Current control in a synchronously rotating dq-reference frame for representation of wind turbine grid-side converter harmonic interaction.
Simulation of harmonics in an offshore WPP and subsequent Fourier analysis of the time- dependent results turned out to be a challenging task. Certain studies suggest accurate results from time-domain simulation of harmonics requires massive parallel processing and efficient modelling of converters [11]. It was decided to work within the frequency domain for this thesis.
Some content from the project’s literature study, although mostly rewritten and restruc- tured, is included to ensure the thesis can be regarded a complete and independent work. As will be shown, the specialisation project’s findings related to emission filter design and converter control also have relevance to the resonance analysis carried out in this thesis.
1.4 Structure
The thesis is organised as follows: Chapter2 presents fundamental theory on harmonics, harmonic resonance and available techniques for harmonic resonance analysis, including HRMA. Chapter3 covers the modelling of power system components for harmonic res- onance analysis. Chapter4 is dedicated to verifying HRMA results which can be found in published papers, hence also validating the implementation of HRMA which has been done from scratch in MATLAB. Important considerations and limitations identified in this process are discussed. In Chapter5, HRMA is applied to a model of a large offshore WPP, based on the Anholt WPP in Denmark. Discussion and Conclusions follow in Chapter6 and Chapter7 respectively. 2 Harmonic Analysis Theory
This chapter presents fundamental concepts and definitions related to harmonics and harmonic analysis. Available methods for harmonic resonance analysis are presented and compared. Theory behind HRMA is presented in detail.
2.1 Harmonics
Any periodic waveform of period T can be expressed as a Fourier Series. This means a function g(t) = g(t + T ) with fundamental frequency f1 can be expressed as
∞ X g(t) = C0 + Ch cos(hf1 + φh) (2.1) h=1 where coefficient C0 is the DC component, Ch is the magnitude and φh is the phase angle of the h-th order integer harmonic component [12].
Waveform distortion can in general be quantified by considering the waveform as a sum of frequency components which are superimposed on the ideal fundamental frequency waveform. In power systems, the dominant distortion components usually appear at frequencies which are integer multiples of the fundamental frequency. Hence, the term harmonic distortion is commonly used as a synonym for waveform distortion, even if the frequency of some components are not integer multiples of the fundamental. The non-integer multiple frequency components are often referred to as inter-harmonics, while non-integer harmonics below the fundamental frequency can be categorized as sub-harmonics [13].
Interest in power quality has surged in recent decades. Increased sensitivity of equipment to voltage quality, and proliferation of power electronic devices in generation, transmis- sion and consumption, are often credited as the main drivers for this. In addition, companies have become less tolerant to equipment malfunction and production process stoppages. Restructuring and deregulation of power markets has also led to an increased need for power quality indicators [3].
4 Chapter 2. Harmonic Analysis Theory 5
Distorted voltage and current waveforms may result in reduced lifetime or failure of sensitive power system components. Voltage distortion affects primarily shunt connected components, ranging from capacitor banks and industrial drives to fluorescent lamps and computer equipment. Current distortion affect primarily the series connected elements, such as transformers [3]. Harmonics can also interfere with control, communication and protective equipment [14].
In relation to offshore WPPs, harmonic distortion is of concern both to the network system operators (i.e. TSO or DSO depending on voltage level at PCC) and to the WPP operators. A system operator is concerned with adhering to regulations on power quality, and therefore may specify connection requirements for WPPs in the form of grid codes. A WPP operator is concerned with grid code compliance, as well as potential damage to equipment and WPP malfunction due to harmonic distortion [9, 13].
There are two main categories of harmonic sources in power systems. The first type of devices are those with non-linear voltage-current relationships, such as iron core reac- tors and transformers. The second category consists of power electronic devices. In a balanced three-phase system, where voltages and current are of equal magnitude with 120 degrees phase shift between phases, only so-called characteristic harmonics are pro- duced by power electronic converters. The characteristic harmonics are defined from converter topology. Because the switching of such devices in steady state is usually synchronised to the fundamental frequency, the characteristic harmonics are also inte- ger harmonics. In reality there is always some degree of asymmetry between phases, causing non-characteristic harmonics to be present as well. These are not related to the converter topology, but are determined by the operating point and control scenario of power electronic converters [12, 14].
Under balanced operating conditions, integer harmonics can be associated to either positive, negative, or zero sequence. Assuming power is supplied in positive sequence, this relationship can be explained by considering the phase shift between phases of harmonics of order h:
Phase a : h · 0◦ Phase b : h · −120◦ Phase c : h · +120◦ (= h · −240◦)
Taking the third harmonic as an example, there is no phase shift between phases, thus the third harmonics are ideally only in the zero sequence. The existence of third order harmonics in the line current is then governed by zero-sequence impedance, which implies the third harmonic current does not flow in the lines of delta or ungrounded systems. Chapter 2. Harmonic Analysis Theory 6
Similarly, it can be verified that in a balanced power system the fifth harmonics are in the negative sequence, the seventh harmonics are in the positive sequence and so on [15].
2.2 Harmonic Resonance
Harmonic resonances influence on harmonic propagation and distortion levels in power systems. Resonances are characterised by energy exchange between capacitive and in- ductive elements in a circuit, hence at least one capacitive and one inductive element must be present in a circuit for resonance to occur. Reactances of capacitors and induc- tors can be expressed respectively as
−1 X = C ωC
XL = ωL where ω has a unit of rad/s and is equal to 2πf where f is frequency in Hz. Hence it is immediately clear that reactances vary with frequency.
Two types of harmonic resonance can occur in electrical circuits: parallel resonance and series resonance. A parallel resonance is exemplified by the circuit in Figure 2.1. The equivalent reactance of the parallel connected capacitor and inductor is given by
1 − ωC · ωL ωL Xeq,parallel = XC ||XL = 1 = 2 (2.2) − ωC + ωL 1 − ω LC
At the resonance frequency the denominator of (2.2) approaches zero. Hence the equiva- lent reactance becomes infinitely high, forcing any harmonic current to flow through the resistor. A parallel resonance thus presents a high impedance to harmonic current close to the resonance frequency. Presence of harmonic current close to the parallel resonance frequency can excite the resonance and cause substantial voltage distortion.
IC+IL
IR IC IL
Ih R C L
Figure 2.1: Parallel resonant circuit. The current (IC + IL) is forced to zero near the resonance frequency, hence any harmonic current Ih must flow through resistor R, which causes voltage distortion. Chapter 2. Harmonic Analysis Theory 7
Ih L R
Vh C
Figure 2.2: Series resonant circuit. At the resonance frequency the equivlaent reac- tance is zero and the flow of current is only a opposed by the resistor. Hence a small harmonic voltage can cause large current distortion.
A series resonance is illustrated by Figure 2.2. The equivalent reactance is given by
1 ω2LC − 1 X = X + X = ωL − = (2.3) eq,series L C ωC ωC
At the resonance frequency the nominator of (2.3) approaches zero. Hence the equivalent reactance goes to zero, and the flow of harmonic current is opposed only by the resistor. A series resonance thus presents a low impedance path for harmonic currents, and can be excited by a small harmonic voltage of frequency close to the resonance.
Comparing (2.2) and (2.3), it is clear that the resonance frequency is the same regardless of the type of resonance, 1 fres = √ [Hz] (2.4) 2π LC
In the per unit system, inductive reactance can be expressed as XL = hX1 where X1 is the fundamental frequency per unit reactance, and h is harmonic number. Similarly,
capacitive reactance can be expressed as 1/(hB1), where B1 is the fundamental frequency capacitive susceptance. The per unit resonance frequency, i.e. the harmonic number of the resonance, can then be expressed as,
r 1 fres,pu = [pu] (2.5) X1B1
A resonance frequency is also the frequency where the inductive reactance and capacitive reactance are equal in magnitude and opposite in sign.
2.3 Methods of Harmonic Analysis
A variety of techniques are available for harmonic analysis in power systems. This thesis focuses primarily on resonance analysis. Because such analysis is usually con- ducted as part of a broader harmonic study, its relation to techniques such as harmonic Chapter 2. Harmonic Analysis Theory 8 power flow and time variant (EMT) analysis is also given some consideration. A general overview of harmonic analysis methods is given in Figure 2.3, dividing them into four main categories. Descriptions of all these forms of harmonic analysis are available in literature [12, 14, 16, 17].
Resonance Analysis Frequency Domain Time Domain Hybrid Techniques
1. Frequency scan 1. Harmonic power 1. Time variant 1. Norton 2. State space analysis flow (current source) analysis (EMT) 2. Transfer fuction 3. HRMA 2. Iterative (voltage 3. Harmonic dependent) harmonic admittance matrix power flow 4. Forward/backward
Figure 2.3: Classification of harmonic analysis methods [16]
Both the frequency scan method and the HRMA are forms of resonance analysis which are strongly related to the admittance matrix and its variation with frequency for the power system of interest. The nodal admittance matrix of a power system at frequency f is of the form
Y11 Y12 ··· Y1i ··· Y1j ··· Y1n Y Y ··· Y ··· Y ··· Y 21 22 2i 2j 2n ...... . . .. . ··· . .. . Yi1 Yi2 ··· Yii ··· Yij ··· Yin [Yf ] = ...... (2.6) ...... Yj1 Yj2 ··· Yji ··· Yjj ··· Yjn ...... ...... Yn1 Yn2 ··· Yni ··· Ynj ··· Ynn
where yji is the mutual admittance between buses j and i, and Yii is the self-admittance of bus i [12]. The admittance matrix defines the relation between bus voltages and currents at any frequency f:
−1 [Vf ] = [Yf ] [If ] (2.7)
[If ] = [Yf ][Vf ] (2.8)
A parallel resonance corresponds to some elements of [Vf ] being large when a per unit current of frequency f is applied at any busbar. A series resonance corresponds to some Chapter 2. Harmonic Analysis Theory 9
elements of [If ] being large when applying a voltage with frequency f at any busbar.
Frequency Scan
The first step in a harmonic study often involves carrying out frequency scans. This method, which is also called frequency sweep or impedance/admittance frequency re- sponse, requires a minimum of input data. It is effective for detection of harmonic resonance, and is also widely used for filter design [14].
By injecting a one per unit current, the measured voltage magnitude and phase angle at bus i corresponds to the driving point impedance at frequency f,
Vi(f) Zii(f) = Ii(f)
This process can be can be repeated at discrete frequencies throughout the range of interest in order to obtain the impedance frequency response. Mathematically, this cor- responds to calculating the the diagonal elements of the impedance matrix, i.e. diagonal elements of [Y ]−1. For example, the driving point impedance at bus i corresponds to the diagonal element at position ii of the impedance matrix [18]. The admittance fre- quency response can be obtained by sweeping the frequency of a one per unit voltage and measuring the current, i.e. Ii(f) Yii(f) = Vi(f)
Impedance and admittance frequency responses are presented as plots showing the mag- nitude and phase angle as functions of frequency. Sharp peaks in impedance magnitude (or valleys in admittance) are indicative of parallel resonance, while sharp valleys in impedance (or peaks in admittance) are indicative of series resonance.
Frequency scans can be performed with phase or sequence components. If all three phases are represented, a set of positive or zero sequence currents may be injected into three phases of a bus to obtain the positive or zero-sequence driving point impedance respectively [12].
Frequency scans are qualitative in nature as they say little about the actual distortion in the power network in the presence of harmonic sources. Moreover they consider only driving impedance, not transfer impedances. For example, consider the circuit of Figure 2.4. It may be verified that equivalent impedance seen from bus 1 at the fifth harmonic frequency is zero, which indicates a series resonance. However, the impedance between bus 1 and bus 2 is equal to −j0.5 at this frequency. Consequently a 5th harmonic current at bus 1 can cause severe voltage distortion at bus 2, with little voltage distortion Chapter 2. Harmonic Analysis Theory 10
Bus 1 Bus 2
j0.1 j0.1
-j1.25 AC
Figure 2.4: Impedances are given in per unit at fundamental frequency. At the fifth harmonic frequency there is a series resonances seen from bus 1. Becuase of the non- zero transfer impedance between bus 1 and bus 2, a fifth harmonic current at bus 1 can drive voltage distortion at bus 2 [17]. at bus 1. Frequency scans are therefore used for quick screening and as a supplement, not as substitute, to more formal harmonic power flow studies. The latter quantify the actual harmonic distortion under given operating conditions [17].
Harmonic Power Flow
The simplest form of harmonic power flow, sometimes referred to as Current Source Method, can be considered an extension of the frequency sweep. For frequency scan, current is injected and voltage measured at one bus at the time. A current source harmonic power flow is carried out at some harmonic frequency by specifying all the injected currents at that frequency in the vector [If ] of (2.7), and then solving for the resulting harmonic voltage vector [Vf ]. This concept can be expanded to include multiple harmonic sources modelled as either current or voltage source. However, this kind of power flow analysis requires the harmonic injections to be predetermined, i.e. given as input to the simulation. More advanced forms of the power flow uses iterative methods and are able to capture the voltage dependency of many harmonic sources [14, 17]. The harmonic power flow is typically repeated at all frequencies of interest, hence estimating the actual harmonic distortion in the presence of harmonic sources. The requirements in terms of input data can be quite extensive. Quantifying actual distortion in this manner is outside the scope of the thesis.
State Space Modal Analysis
In the field of power system analysis, modal analysis is best recognized as a technique for dynamic small signal stability analysis. The state space representation of a linear Chapter 2. Harmonic Analysis Theory 11 system is written as x˙ = Ax + Bu (2.9) where x is a vector of state variables referred to as state vector, A is the state matrix, B is the input matrix, and u is the external excitation input vector. The state space model of a linear electrical circuit can be formulated by selecting inductor currents and capacitor voltages as state variables. Solving the differential equations yields a time domain transient response given by
Z t A(t−t0) A(t−τ) x(t) = e x(t0) + e Bu(τ)dτ (2.10) t0
where eA must be determined using eigendecomposition.
Eigenvalues of the state matrix A are the system modes. The number of modes is equal to the rank of the state matrix. Complex eigenvalues can only exist as complex conjugate pairs, and such a pair is often referred to as an oscillatory mode. The complex modes are of the form
λA = σA ± jωA (2.11)
where σA is a mode damping coefficient with unit 1/s, and ωA describes the mode damped frequency with unit rad/s. A 2nd order system has 2 eigenvalues, and assuming they are a complex conjugate pair as given by (2.11), a small disturbance will lead to oscillation of the state variables with a frequency determined by ωA and a damping time-constant of 1/σA,
−σAt x(t) = K1e cos(ωAt − φ) (2.12)
where K1 is a constant and φ represents a phase lag. For a higher order system with more than one oscillatory mode, the small disturbance response will be a linear combination of oscillatory responses determined by the modes [6].
As seen, the modes of the state matrix characterise the transient response of the system. The modes also indicate the stability margin of a system when subject to a small distur- bance. This kind of modal analysis forms the basis for small signal stability analysis [19].
If there is a disturbance with a frequency close the ωA of an oscillatory mode, the system transient response tends to be strong. From this perspective, the oscillatory modes can
also be used to characterise resonance frequencies. However, the natural frequency ωA is the same as impedance resonance frequency only for small values of σA [6]. Chapter 2. Harmonic Analysis Theory 12
2.4 Harmonic Resonance Mode Analysis (HRMA)
There are many applications of modal decomposition and eigenvalue analysis in the field of power system analysis. Eigenvalue analysis can be applied when the system description is based on linear differential equations as well as when the system descrip- tion is based on algebraic equations. The application in state space analysis has already been mentioned (linear differential equations). Another well-known application of modal decomposition is in the representation of three-phase systems using symmetrical com- ponents (algebraic equations). Application is also found in voltage stability analysis, where eigenvalues of the Jacobian matrix (algebraic equations) can be used to quantify the voltage stability margin [5, 19]. HRMA is a form of eigenvalue analysis applied to a system description based on algebraic equations.
It has been found that parallel resonance phenomena is associated to frequencies at which the nodal admittance matrix approaches singularity, which is equivalent to an eigenvalue of the matrix approaching zero. Hence it has been suggested that eigenvalues and associated eigenvectors at such critical frequencies contain information about parallel resonances which can enhance understanding of resonance phenomena and assist in resonance mitigation. The idea of using a modal decomposition of the Y-bus matrix for harmonic resonance analysis was introduced by Xu et al. [5]. The concept was expanded upon by Huang et al. [6], Cui et al. [7, 20], and Hu et al. [8]. Theory and notation regarding HRMA presented in this thesis originates primarily from these papers. Application examples of HRMA can be found in the aforementioned papers, as well as in papers dedicated to exploring particular types of networks, such as distribution networks [21], offshore collection and transmission networks for wind power plants [22– 24] and wave power plants [25].
Eigendecomposition
The term right eigenvector refers to a column vector t which is placed to the right of matrix A in the defining equation At = λt (2.13) where λ is the eigenvalue associated with the eigenvector. The term left eigenvector refers to a row vector l which must be placed to the left of the matrix A in the defining equation lA = lλ (2.14)
An admittance matrix is symmetric as long as the phase shifting effect of transformers is neglected. The non-conjugate transpose og an admittance matrix Y is therefore equal Chapter 2. Harmonic Analysis Theory 13 to itself. It is therefore easily shown that the (non-conjugate) transpose of any right eigenvector t is equal to a left eigenvector l:
(Y t)T = (tλ)T ⇒ tT Y = tT λ (2.15)
Eigendecomposition of an admittance matrix can thus be written as
[Y ] = [L][Λ][T ] (2.16) where [Λ] is the diagonal eigenvalue matrix, [L] consists of columns of eigenvectors, and [T ] consists of rows of eigenvectors, i.e
. . . . . ··· . T [L] = [T ] = l1 l2 ··· ln (2.17) . . . . . ··· .
One can conclude that [L] = [T ]−1 = [T ]T and vice versa.
The symmetric admittance matrix used in power system analysis is defined by
−1 [Vf ] = [Yf ] [If ](2.7 revisited) where [Vf ] and [If ] are the nodal voltage and current vectors respectively at frequency f. The subscript f is dropped for simplicity in the following. Substituting (2.16) into (2.7) yields, [V ] = [T ]−1[Λ]−1[L]−1[I] = [L][Λ]−1[T ][I] or [T ][V ] = [Λ]−1[T ][I] (2.18)
Defining the modal voltage vector [U] = [T ][V ] and modal current vector [J] = [T ][I], (2.18) can be simplified to [U] = [Λ]−1[J] which in expanded form is given by
−1 U1 λ1 0 0 0 J1 −1 U2 0 λ 0 0 J2 2 . = . . (2.19) . 0 0 .. 0 . −1 Un 0 0 0 λn Jn Chapter 2. Harmonic Analysis Theory 14
The kth modal impedance is defined as
1 Zm,k = (2.20) λk
A small eigenvalue λk will result in a large modal impedance Zm,k, in which case a small modal current Jk at will lead to a large modal voltage Uk. This defines a parallel resonance in the modal domain. If the system has no damping, the admittance matrix becomes singular at resonance frequencies, which is equivalent to having an eigenvalue of zero, or an infinite modal impedance. There is no cross-coupling between the n equations given by (2.19), hence the excitation of one mode only affects a single modal voltage.
One important consideration is that the modal problem formulation of (2.19) is only suitable for detection and analysis of parallel resonances. There is no direct link between the eigenvalues of the admittance matrix and series resonances. Im many cases the behaviour of harmonic sources closely resemble that of a current sources, hence parallel resonances are most critical. However, there is an alternative system description which can be used to analyse series resonances. Such a description is given by
[E] = [Zloop][Iloop] (2.21) where [E] is the loop voltage vector, [Zloop] is the loop impedance matrix, and [Iloop] is the loop current vector. The eigendecomposition of [Zloop] can be obtained in a similar manner to the Y -bus matrix for analysis of series resonance [5]. However, HRMA for parallel resonance analysis has been developed much further than for series resonance analysis. Only the formulation suitable for analysis of parallel resonances is considered further in this thesis.
Modal Participation Factors
Let us assume the eigenvalue λ1 of (2.19) to be very small at some frequency associated to a parallel resonance. The modal current vector is related to the current vector [I] by
[J] = [T ][I]. The modal current J1 is therefore given by
J1 = T11I1 + T12I2 + T13I3 + ··· + T1nIn (2.22) where T1,n is the n-th element of the eigenvector associated with λ1. This means the eigenvector associated with λ1 determines controllability, i.e. contribution of each nodal current to the modal current J1. The nodal bus voltages are related to the modal Chapter 2. Harmonic Analysis Theory 15 voltages by [V ] = [L][U], or
V1 L11 L12 L1n L11 V2 L21 L22 L2n L21 . = . U1 + . U2 + ··· + . Un ≈ . U1 (2.23) . . . . . Vn Ln1 Ln2 Lnn Ln1 where Ln,1 is the n-th element of the eigenvector associated to λ1. The approximation of (2.23) holds because U1 has a much larger value than the other modal voltages at the resonance associated to the small value of λ1. The eigenvector associated to λ1 thus also determines observability, i.e. reveals how observable the modal voltage U1 is in each of the nodal voltages. The left and right eigenvectors associated to an eigenvalue are non-conjugate transpose of each other, hence the bus that has highest controllability of a modal resonance also has the highest observability of the same modal resonance.
The index of participation factor combines the observability and controllability of modes.
−1 λ1 0 0 0 0 λ−1 0 0 −1 2 [V ] = [L][Λ] [T ][I] = [L] . [T ][I] 0 0 .. 0 −1 0 0 0 λn −1 L11 L12 ··· L1n λ1 0 0 0 T11 T12 ··· T1n L L ··· L 0 0 0 0 T T ··· T 21 22 2n 21 22 2n ≈ [I] ············ 0 0 0 0 ············ Ln1 Ln2 ··· Lnn 0 0 0 0 Tn1 Tn2 ··· Tnn L11T11 L11T12 ··· L11T1n I1 L T L T ··· L T I −1 21 11 21 12 21 1n 2 = λ1 (2.24) ············ ··· Ln1T11 Ln1T12 ··· Ln1T1n In
The approximation is made possible because the critical mode impedance, 1/λ1, is much larger than the other modal impedances. The diagonal elements of the above matrix are the nodal participation factors in the critical mode. They can be defined as PFik =
LikTki, where i is the bus number and k is the modal impedance number.
The magnitude of a modal impedance is only meaningful in a physical sense when considered in conjunction with participation factors. The driving point impedances seen from bus i at some frequency can be expressed via participation factors and eigenvalues Chapter 2. Harmonic Analysis Theory 16 at that frequency:
1 1 1 1 Zii = PFi1 + PFi2 + PFi3 + ··· + PFin λ1 λ2 λ3 λn
= Zm,1PFi1 + Zm,2PFi2 + Zm,3PFi3 + ··· + Zm,nPFin (2.25)
Weighted Participation Factor (WPF) can be defined as:
WPFik = Zm,kPFik (2.26) where i is bus number and k is modal impedance number. Expressing (2.25) in terms of WPFs yields,
Zii = WPFi1 + WPFi2 + WPFi3 + ··· + WPFin (2.27)
A resonance detected from driving point impedance scan at bus i can hence be considered a result of exciting one or more modal resonance modes.
Modal Impedance Sensitivity Index
In addition to participation factors, it has been suggested that the involvement of net- work components in modal resonances can be quantified using sensitivity indices. These indices indicate how much a modal resonance moves in the impedance-frequency plane following a small change in some network parameter α. The movement of a modal resonance in the three-dimensional f-Z-α domain following a small change of α is illus- trated in Figure 2.5. The modal impedance sensitivity index quantifies vertical move- ment, while the frequency sensitivity index quantifies the horizontal movement in the frequency-impedance plane.
The sensitivity of the k-th eigenvalue of the Y -bus admittance matrix to some parameter α can be expressed as ∂λ ∂Y k = t l (2.28) ∂α k ∂α k where tk is the k-th row eigenvector of [T ] and lk is the k-th column eigenvector of [L]. This equation suggests the sensitivity of an eigenvalue with respect to some parameter α can be obtained to evaluating the sensitivity of the admittance matrix to the same parameter.
Let’s initially consider the evaluation of (2.28) with respect to a shunt admittance at ∂Y bus i given by ysh,i. The i-th diagonal element of ∂α takes on value 1, with all other Chapter 2. Harmonic Analysis Theory 17
Z [pu]
ΔZm
Zm(α)
f [pu]
Δα
fm(α) Δfm
α [%]
Figure 2.5: The small change ∆α of some network parameter α causes a the modal resonance peak to move a vertical distance ∆Zm and a horisontal distance ∆fm in the three dimensional f-Z-α domain [7]. elements being zero. For a shunt admittance at bus 2, this yields
0 0 ··· 0 ∂Y 0 1 ··· 0 = . . . . ∂ysh,2 . . .. . 0 0 0 0
Now consider evaluation of (2.28) with respect to a series admittance between bus i and ∂Y bus j. Four elements of ∂α take in non-zero values, e.g. for a series admittance between bus 1 and bus 2, 1 −1 ··· 0 ∂Y −1 1 ··· 0 = . . . . ∂yse,1−2 . . .. . 0 0 0 0 Chapter 2. Harmonic Analysis Theory 18
Defining the sensitivity matrix with respect to eigenvalue λk,
. . ············ . . . . .. . h i . ······ Sk = lktk = ······ T ··· = (2.29) ki . . Ljk . . LjkTki = sk,ji ··· . . . . . . ······
The eigenvalue sensitivity of mode k to a shunt branch at bus i is given by
∂λk ∂Y ∂Y ∂Yii = tk lk = tk lk = sk,ii (2.30) ∂ysh ∂ysh ∂Yii ∂ysh,i
Hence the sensitivity of eigenvalue k with respect to shunt admittance branches at defined buses are given by the diagonal elements of the sensitivity matrix Sk. The diagonal elements of this matrix are in fact the same as bus participation factors. The eigenvalue sensitivity with respect to a series admittance branch between bus i and bus j can be expressed as
∂λk ∂Y = tk lk = sk,ii + sk,jj − sk,ij − sk,ji (2.31) ∂yse,ij ∂yse,ij
Admittances ysh and yse and eigenvalues λk are complex numbers. However, it is the sensitivity of |λk|, i.e. the sensitivity of the absolute value which is of primary interest. Moreover, the derivative with respect to a complex number is somewhat confusing. Hence it is desirable to obtain the derivative of eigenvalue magnitude with respect to real and imaginary part of admittance. This can be done by defining
λk = λr + jλi,Sk = Sr + jSi 2 y = Gbr + jBbr F = |λk| where Sk is either given by (2.30) or (2.31) depending on whether the branch in question is shunt or series connected. A change of ∆Bbr results in a change ∆F which is given by
2 2 2 2 ∆F = |λk + ∆λk| − |λk| = |(λr + jλi) + (Sr + jSi)(j∆Bbr)| − |(λr + jλi)| 2 2 2 2 = (λr − Si∆Bbr) + (λi − Sr∆Bbr) − (λr + λi ) 2 2 2 = 2(Srλi − Siλr)∆Bbr + (Sr + Si )(∆Bbr)
Taking only the first order term and letting ∆Bbr → 0,
∂F = 2(Srλi − Siλr) ∂Bbr Chapter 2. Harmonic Analysis Theory 19
Similarly ∂F = 2(Srλr + Siλi) ∂Gbr
Converting back from F to |λk| results in
∂|λk| Srλr + Siλi = q = µ (2.32) ∂Gbr 2 2 λr + λi
∂|λk| Srλi − Siλr = q = ν (2.33) ∂Bbr 2 2 λr + λi
Lets assume all branches in a harmonic study are fully representable by a series RLC branch with linear components (i.e. constant value of R, L and C). Any branch is then fully defined from its fundamental frequency value or resistance R1, inductive reactance
X1 and capacitive susceptance B1. The conductance and susceptance of the branch can be expressed from these properties.
R G = 1 (2.34) br 2 R2 + hX − 1 1 1 hB1 1 − hX1 B = hB1 (2.35) br 2 R2 + hX − 1 1 1 hB1
In order to find the modal impedance sensitivity with respect to individual components of the RLC branch, one can employ the chain derivative rule.
∂|λ | ∂|λ | ∂G ∂|λ | ∂B ∂G ∂B k = k br + k br = µ br + ν br (2.36) ∂α ∂Gbr ∂α ∂Bbr ∂α ∂α ∂α where α can be R1, X1 or B1 [7,8]. Mathematical expressions for the derivative of
(2.34) and (2.34) with respect to X1 and B1 have been obtained using Maple software from Maplesoft. These expressions are included in AppendixB.
The modal impedance sensitivity index can be expressed as
∂Zm,k ∂Zm,k ∂|λk| 1 ∂|λk| = = − 2 (2.37) ∂α ∂|λk| ∂α |λk| ∂α
The modal impedance sensitivity can be normalised for easier comparison of network components with different units. Dropping the subscript k for generality,
∂Z ∂Zm ∂|λ| α % m = Zm = − · (2.38) ∂α norm ∂α ∂α |λ| 100 % α Chapter 2. Harmonic Analysis Theory 20
The unit of the normalised sensitivity index used in this thesis is percent change of modal impedance per 100 percent change of network parameter.
Modal Frequency Sensitivity Index
An important realisation is that the magnitude of a modal impedance near resonance is mainly influenced by resistive and not reactive components of participating branches. For example, in the simple circuit of Figure 2.1, the impedance is purely resistive at the resonance frequency, 1 1 Zm = = 1 1 = R λ R + j(ωC − ωL ) i.e. the (modal) impedance magnitude is purely a function of resistance R. On the contrary the resonance frequency of the circuit in Figure 2.1 is determined exclusively by the reactive components L and C, as given by (2.4). A frequency sensitivity index, ∂f/∂α which quantifies the change in resonance frequency with respect to a network component has therefore also been suggested in literature. Such an index is possible to develop, though its calculation becomes more complicated than the modal impedance sensitivity index.
Derivation of the frequency sensitivity index begins with defining a resonance peak of some modal impedance frequency response at frequency fres,1,
∂Z m = 0 (2.39) ∂f f=fres,1 α=α1
The linear Taylor expansion of this expression around the modal impedance peak is given by
2 2 ∂Zm ∂Zm ∂ Zm ∂ Zm (fres,1 + ∆f, α1 + ∆α) = + ∆f + ∆α (2.40) ∂f ∂f ∂2f ∂f∂α f=fres,1 α=α1
A small change ∆α in the magnitude of α will cause a shift of the resonance frequency. This new modal impedance peak can be defined as
∂Z m = 0 (2.41) ∂f f=fres,1+∆fres α=α1+∆α
Expressing (2.41) with the linear Taylor expansion of (2.40) and simplifying with (2.39) yields 2 2 ∂ Zm ∂ |λ| ∂fres ∆fres ∂f∂α ∂f∂α = = − 2 = − 2 (2.42) ∂α ∆α ∂ Zm ∂ |λ| ∂2f ∂2f Chapter 2. Harmonic Analysis Theory 21
The denominator of (2.42), i.e. the second-order derivative of the eigenvalue with respect to frequency can be calculated numerically during the modal impedance scan process. Considering the eigenvalues are calculated at discrete frequencies with step length of ∆f, ∂2|λ| 1 = (|λ| + |λ| − 2|λ| ) (2.43) ∂2f 4∆f 2 f+2∆f f−2∆f f where subscripts denote at which discrete frequency the eigenvalue is obtained relative to the current discrete frequency f.
The nominator of (2.42) can be approximated from the derivative with respect to fre- quency of ∂|λ| ∂|λ| ∂G ∂|λ| ∂B = br + br (2.36 revisited) ∂α ∂Gbr ∂α ∂Bbr ∂α which yields ∂2|λ| ∂µ ∂G ∂2G ∂ν ∂B ∂2B = br + µ br + br + ν br (2.44) ∂f∂α ∂f ∂α ∂f∂α ∂f ∂α ∂f∂α Expressions µ and ν are defined in (2.32) and (2.33) respectively. The terms ∂µ/∂f and ∂ν/∂f can be obtained numerically during the modal impedance scan process,
∂µ 1 ∂ν 1 = (µ| − µ| ) , = (ν| − ν| ) (2.45) ∂f 2∆f f+∆f f−∆f ∂f 2∆f f+∆f f−∆f
2 2 The remaining terms in (2.44), i.e. ∂Gbr/∂α, ∂ Gbr/∂α∂f, ∂Bbr/∂α and ∂ Bbr/∂α∂f must be explicitly calculated depending on α and the kind if branch which is to be analysed. Again, many branches that are suitable for inclusion in a harmonic study can
be fully represented by an RLC branch, in which case Gbr and Bbr can be expressed as in (2.34) and (2.35). Expressions for the remaining terms given the RLC branch assumption have been obtained using Maple, and are included in AppendixB.
The modal frequency sensitivity index can be normalised in order to facilitate comparison of network components with different units. The normalised modal frequency sensitivity index utilised in this thesis is defined as the per unit change of resonance frequency to a 100 percent change in parameter α. It can be expressed as
∂f ∂f ∂f h pu i res = res = res · α (2.46) ∂α norm ∂α ∂α 100 % α and is achieved simply by multiplication with the value α of the network component [7,8]. Chapter 2. Harmonic Analysis Theory 22
2.5 Comparing Modal Analysis Techniques
As seen in the previous sections, both the state A matrix and the Y -bus admittance matrix can be used to analyse harmonic resonances. The oscillatory modes of A are of the form
λA = σA ± jωA (2.11 revisited)
The resonance modes in HRMA are each associated to a resonance frequency fres, and a modal impedance Zm. A comprehensive comparison of resonance analysis using HRMA and state state space analysis was carried out by Huang et al. [6]. It was shown that the resonance frequencies obtained from the state matrix A, termed ωA and expressed
in rad/s, coincide with those frequencies obtained from the Y -bus matrix, given by ωY in rad/s, only if damping index σA is zero or very close to zero. In addition, there is no simple relationship between the damping index of σA and the modal impedance Zm. Hence there is no straightforward correspondence between the oscillatory modes of A and the resonance modes of the admittance matrix Y .
Distinct characteristics which separate the state space modal analysis technique and HRMA include:
• The modes of the A have a unit of 1/s and characterise the time-domain transient response of a network. On the contrary, modal variables of the Y -bus matrix have the unit of impedance (Ω or pu). The resonance modes of the admittance matrix characterise the electrical circuit in frequency domain.
• The admittance matrix is a nodal system description. This results in eigenvectors which carry additional information about nodal participation in resonance modes. A state space model is not a nodal description, hence the eigenvectors of A do not offer information about nodal participation in a resonance.
• The number of modes given by the state matrix is equal to the rank of A, and the oscillatory modes appear as discrete points in the frequency domain. The number of modal variables in HRMA is equal to the rank of the admittance matrix, and these variables are defined at all frequencies.
• Sensitivity indices for the oscillatory modes of A with respect to network compo-
nents can be calculated as dσA/dα and dωA/dα. The sensitivity indices for modes of the Y -bus matrix are more elaborate to calculate, though it can be managed as shown in previous sections, or they can be estimated numerically.
In addition, there are modelling limitations with the state-space approach. In partic- ular, network elements with distributed and frequency dependent parameters, such as Chapter 2. Harmonic Analysis Theory 23 transmission lines and cables, are not easily included in the state-space network model, but can be highly important for accurate assessment of harmonic resonance. The repre- sentation of such impedances in the state matrix requires synthesizing equivalent RLC networks using special techniques such as vector fitting [26]. On the contrary, the admit- tance approach used in HRMA allows for recalculating the parameters of admittances at each discrete frequency, which makes it much easier to handle non-linearities and frequency dependency. 3 Modelling for Harmonic Resonance Studies
This chapter presents the modelling of power system components which are later in- cluded in HRMA of specific cases. The extent and level of detail in modelling of net- work components for harmonic studies depends on the analysis technique to be used, problem formulation, frequency range of interest, required accuracy of results and so on. Most HRMA presented in published papers assume linearity of network parameters, i.e. constant values of R, L and C over the frequency range of interest. This chapter present linear models as well as some effects which can cause non-linearities.
3.1 Overhead Lines and Cables
Multiphase cables and transmission lines can be represented with coupled π-equivalent circuits. Most power system analysis software aimed at harmonic and/or electromag- netic transient analysis contain modules for calculation of parameters based on physical arrangements specified by the user [15, 27]. Once the parameters of a line or cable are known, the admittance or impedance matrix describing phase to ground and mutual ad- mittances/impedances can be constructed. For balanced studies, coupled π-equivalent models can be simplified into separate positive, negative and zero sequence π-equivalents as shown in Figure 3.1.
Two important considerations related to the electrical parameters of lines and cables are:
• The distribution of parameter along the line/cable length which yields so-called long line effects
• The frequency dependency of parameters due primarily to skin effect and earth return
If long line and frequency dependent effects of the electrical parameters are neglected, the positive sequence model parameters are given by the unit-length parameters multiplied
24 Chapter 3. Modelling for harmonic resonance studies 25
IS IR Z
VS Y1 Y2 VR
Figure 3.1: π-equivalent line/cable model. Subscripts S and R denote sending and receiving end respectively. For steady state analysis in frequency domain, the lumped parameters Z, Y1 and Y2 can be adjusted with long line correction factors to obtain exact representation of long line effects at a particular frequency. by length. The total shunt admittance is simply divided equally between the two lumped shunt admittances. This is known as the nominal π-equivalent model.
Znominal = R1 + jhX1 (3.1) Y1,nominal = Y2,nominal = jhB1/2
where R1, X1 and B1 are the fundamental frequency values of total length resistance, inductive reactance and capacitive susceptance respectively.
The long line effects can be accounted for by including the long line hyperbolic correction factors. sinh(γl) Z = Z sinh(γl) = Z exact 0 nominal (γl) (3.2) 1 γl tanh(γl) Y1,exact = Y2,exact = tanh( ) = Ynominal Z0 2 (γl/2) where
√ γ = zy rz Z = 0 y and z is the unit-length series impedance, y the unit-length shunt admittance, and l is the total length. The hyperbolic correction factors yield an exact representation of the long-line effects from the cable terminals at the frequency they are calculated for [28].
Skin effect forces alternating currents towards a conductors perimeter, resulting in cur- rent density which tapers off from conductor surface towards its centre. Frequency dependency of parameters due to skin and proximity effects is typically accounted for Chapter 3. Modelling for harmonic resonance studies 26 by using frequency dependent correction factors. All line and cable parameters are fre- quency dependent, but the effect is most prominent in series resistance. In this thesis, the formula suggested by the National Grid Company is used for series resistance,
√ R = R1(0.187 + 0.532 h, h > 2.35) (3.3) where R1 is fundamental frequency resistance, and h is harmonic order [12]. The cor- rected value of resistance R can then be applied in (3.2).
For calculations at power frequency involving overhead lines, it is often considered ac- ceptable to neglect the shunt admittance for lines shorter than 80 km. A model consisting simply of the series impedances is therefore also known as the short line model. Similarly, the nominal π-equivalent is generally considered acceptable for line lengths up to 250 km at power frequency, hence it is known as the medium line model. Cables generally have much larger shunt capacitances than overhead lines, hence long line effects should be taken into account for much shorter cables. The long line effects are also much more prominent with higher frequencies, making short and medium length approximations less suited for harmonic studies [14, 29].
3.2 Transformers
The short circuit impedance, magnetizing characteristics and vector group (winding connections), are the main characteristics of transformers which impacts on harmonic flows. Transformers are typically modelled in positive sequence (and negative sequence) as T-equivalent circuits like the one shown in Figure 3.2.
R1=RS/2 L1=LS/2 L2=LS/2 R2=RS/2
GB XM
Figure 3.2: Transformer T model. Phase shifter due to vector group and harmonic current source for representation of transformer generated harmonics not shown. Chapter 3. Modelling for harmonic resonance studies 27
For simplified resonance studies, the magnetizing branch may often be ignored, modelling the transformer only by its short-circuit impedance. If any kind of harmonic power flow, current source or iterative, involving more than a single harmonic source is to be carried out, transformers phase shift must be taken into account. The phase shift in transformers typically yields a high impedance in zero sequence, ideally resulting in cancellation of zero-sequence harmonics. Modelling of non-linear effects in transformers due to saturation can be achieved by having a voltage dependent current source in parallel with the magnetizing branch, but this is usually only included when transformer- generated harmonics are the primary concern [14, 29].
3.3 Aggregate Loads
An aggregate load refers to group of buses that are treated as a single network element for harmonic analysis. Typical aggregate loads are feeders seen from the substation of a distribution system, or a larger customer seen from the point of coupling. Such loads may provide the majority of damping component in distribution networks and also affect resonance conditions at higher frequencies. If there are harmonic sources of concern in these loads, they should not be treated as an aggregate load.
R=U2/P
R=U2/P X=U2/(hQ) X=(hR)/(6.7(Q/P)-0.74)
X=0.073hR
Conventional load model CIGRE load model
Figure 3.3: Aggregate load models for harmonic studies. Parameters are estimated using rated network voltages and stady state power flow.
A load model represents the impedance frequency response of the aggregate loads in a simplified way. The impedance response depends on the type of loads as well as the line or cable which feeds the loads. Consequently, the applicability of general aggregate load Chapter 3. Modelling for harmonic resonance studies 28 models is quite limited. Nevertheless, suggested representations of aggregate loads are found in literature [29]. Two typical models found in literature are shown in Figure 3.3.
3.4 VSC Grid-side Harmonic Emission Filters
Voltage Source Converters (VSC) are utilised in full-scale converter wind turbines. When emissions and grid interaction of such converters are of concern, they need to be modelled accurately for harmonic studies.
The grid-side terminals of a WTG with full-scale converter is typically connected to the MV collector system via a coupling reactor with inductance Lmr and a step-up transformer with short circuit inductance Ltr. In addition there are shunt harmonic filters between the reactor and transformer as shown in Figure 3.4. The shunt harmonic filters are capacitive at fundamental frequency, and although the shunt harmonic filter topology can vary, such a combination of inductor Lmr, inductor Ltr and shunt harmonic filter can be said to constitute an LCL filter.
L2=Ltr L1=Lmr IL1 IL2
Harmonic Vconv Shunt Vgrid Filter(s)
Figure 3.4: Basic filtering structure commonly used for grid-tied VSCs. The shunt harmonic filter, generally capcitive at fundametal frequency, can take on different topologies. Overall layout can be refered to as LCL filter.
There are a number of constraints in designing filtering solutions for grid connected converters. The filters must be designed not only to ensure compliance with relevant grid codes, but must also facilitate stable operation of the converter controllers. In the multi- megawatt power range, the capability of semi-conductor switches can further constrain parameters such as switching frequency and maximum DC voltage [30]. Moreover it is desirable to minimize energy losses, volume, weight and cost of the overall filtering solution. Due to increasing numbers of grid connected VSCs in modern power systems, Chapter 3. Modelling for harmonic resonance studies 29 the optimisation of emission filters is a topic which is receiving attention in research [9, 10].
3.5 Harmonic Sources/Full-Scale VSC Wind Turbines
The classic approach for harmonic studies in frequency domain is to model harmonic sources as fixed ideal current sources. For many harmonic sources, such as line-commutated HVDC converters, this approximation is fairly accurate. However, the emission from other harmonic sources can be highly dependent on existing voltage distortion and controller interaction, in which case the ideal current source method can be insuffi- cient [14, 17].
State of the art WTG harmonic models consist of a number of Norton or Thevenin equivalents with Norton or Thevenin impedance specified for every harmonic frequency of interest. Such models can be derived by the turbine manufacturer, which has full access to the converter control structure and settings. These models facilitate analysis of controller impact on resonances as well as analysis of countermeasures such as active filtering via the converter controller [31, 32].
Converter Control Interaction
In order to investigate the interaction between converter control and harmonic reso- nances, there is a need to represent the grid-side converter in frequency domain as seen from elsewhere in the network. A VSC with simplified current control and admittance as seen from the AC-side terminals of the converter is shown in Figure 3.5.
Main Shunt Turbine Grid reactor filter transformer impedance Ig(s) Y(s) Zd(s)
Vg(s) Zmr(s) Z (s) V (s) tr I*(s) g Gi(s) Zsf(s) Znet(s)
Ig(s)
Figure 3.5: Grid-tied VSC (of WTG) with indication of current control loops and impedance as seen from converter terminals [33]. Chapter 3. Modelling for harmonic resonance studies 30
Zd(s)
Vg(s) I*(s) V*(s) Ig(s) Gi(s) Y(s)
Ig(s)
Figure 3.6: Current control loops with current feed-back and voltage feed-forward. The voltage feed-forward enhances disturbance rejection of the converter [33].
A typical current control structure with current feed-back and voltage feed-forward is shown in Figure 3.6. The open-loop transfer function can be obtained from the block diagram, Y (s) Hol(s) = Gi(s) (3.4) 1 − Y (s)Zd(s)
Gi(s) is the transfer function of the current control, including any harmonic compensa- tion, filtering and decoupling. The other transfer functions are defined as
Zd(s) = (Ztr(s) + Znet(s))kZsf (s) (3.5)
1 Y (s) = (3.6) Zmr(s) + Zd(s) The voltage feed forward can be combined with Y (s) to obtain an equivalent system transfer function,
Y (s) 1 1 Gsys(s) = = = (3.7) 1 − Zd(s)Y (s) Zmr(s) Rmr(1 + sτmr)
where τmr = Lmr/Rmr is the time constant of the main reactor.
The stability of the current control loop can be evaluated based on the open loop transfer function (3.4). For such purposes the delay due to Pulse Width Modulation (PWM) and discrete sampling should also be represented. The stability can be evaluated using Nyquist stability criterion, and different operation conditions of the external grid can
be considered by varying the impedance of Znet(s)[33].
From the current control structure it is possible to derive an average harmonic model which can represent the grid-tied VSC as seen from elsewhere in the network. Deviation from the ideal sinusoidal voltage waveform due to forcing external to the converter, i.e.
background harmonic voltage, can be considered as a disturbance Vg(s) acting on the process as shown in Figure 3.7[33]. Chapter 3. Modelling for harmonic resonance studies 31
Vg(s)
I*(s) Ig(s) Gi(s) Gsys(s)
Ig(s)
Figure 3.7: VSC current control loop with background harmonic voltage modelled as disturbance.
Neglecting PWM and sampling delays, the following relation can be obtained:
∗ [(I (s) − Ig(s))Gi(s) − Vg] Gsys(s) = Ig(s) (3.8)
Solving for current Ig(s) yields
∗ Ig(s) = Gc(s)I (s) − Yeq(s)Vg(s) (3.9) where
Gsys(s) Gi(s)Gsys(s) Yeq(s) = Gc(s) = Gi(s)Yeq(s) = (3.10) 1 + Gi(s)Gsys(s) 1 + Gi(s)Gsys(s)
Expressing the system transfer function Gsys(s) via the main reactor impedance Zmr(s),
Yeq(s) and Gc(s) can alternatively be written as
1 Gi(s) Yeq(s) = Gc(s) = Gi(s)Yeq(s) = (3.11) Zmr(s) + Gi(s) Zmr(s) + Gi(s)
Equation (3.9) can be visualised as the Norton equivalent harmonic model in Fig- ure 3.8[34–36]. The Norton equivalent can easily be converted to a Thevenin equivalent if necessary.
The harmonic model of (3.9) does not represent switching harmonics of the converter itself, but rather represent the converter as seen from elsewhere in the network. Qual- itatively, the current controller tends to have high open loop gain within the controller bandwidth, i.e. |Gc(s)| ≈1 at lower frequencies. Above the controller bandwidth the current controller tends to have low gain, i.e. |Gc(s)| ≈0, hence the converter acts like as passive component with admittance Yeq(s)[36]. Chapter 3. Modelling for harmonic resonance studies 32
Converter harmonic model WTG filter and transformer Grid equivalent
Ig(s)
Ztr(s) Znet(s) Gc(s)I*(s)
Yeq(s) Vg(s) Zsf(s)
Figure 3.8: Norton equivalent harmonic model together with WTG filter, transformer and grid equivalent.
The current control loops can be implemented in a stationary or a synchronous reference frame. Regulating currents in stationary abc or αβ reference frames with Proportional Integral (PI) regulators in theory should be effective. The transfer function of a PI- regulator is given by 1 Gi,P I (s) = Kp 1 + (3.12) Tis However, PI control of stationary reference frame currents is hampered by PWM and sampling delays which limits the maximum controller gain and hence leads to steady state tracking error. This issue is commonly addressed by transforming the phase cur- rents into a synchronously rotating dq-reference frame and controlling the d- and q-axis currents. PI current regulators in synchronous dq-reference frame is essentially equiv- alent to a Proportional Resonant (PR) regulator in a stationary reference frame. The PR-regulator transfer function is given by
s Gi,P R(s) = Kp 1 + 2 2 (3.13) Ti(s + ω0)
This (ideal) resonant regulator provides infinite gain at the tuned frequency ω0, which is set equal to the fundamental frequency. Hence steady state tracking error can be eliminated. As frequency increases above the fundamental, the transfer function response of (3.13) approaches that of (3.12)[34]. It therefore seems reasonable to assume PI- regulator behaviour for harmonic frequencies.
Let’s now approximate the positive sequence current to be regulated by a conventional PI-regulator. Any specific measures for harmonic rejection, filtering and decoupling in
Gi(s) are neglected. The equivalent impedance of the converter can from (3.11) be Chapter 3. Modelling for harmonic resonance studies 33 written as,
1 Zeq(s) = = Zmr(s) + Gi(s) Yeq(s) 1 = Rmr(1 + sτmr) + Kp 1 + (3.14) Tis
It should be appreciated that the pole of the PI-regulator gives it a capacitive behaviour for the lower frequency range (i.e. below the controller knee-point frequency). Replacing the complex frequency s with jω yields an impedance given by
Kp Zeq(ω) = Rmr + jωLmr + Kp − j ωTi 1 = (Rmr + Kp) + j ωLmr − (3.15) ω Ti Kp
When selecting regulator parameters Kp and Ti a first order approximation for PWM and sampling delays is often taken into account. Its time constant can be approximated as 1.5 Ta = 1.5Tsw = fsw where fsw is the PWM switching frequency. Delays of Tsw and 0.5Tsw are due to sampling and PWM respectively. The system pole represented by τmr is much larger than the one due to delays. The Modulus Optimum Criterion suggests the zero introduced by the PI-regulator can be set to a value which cancels the slow process pole represented by τmr. The value of Kp can further be chosen to achieve optimal flatness of the closed loop second order response [9]. The resulting PI-regulator settings are given by
Lmr Kp = ; Ti = τmr (3.16) 2Ta
3.6 Wind Power Plant Model Aggregation
There are several techniques for aggregation of WPP array collector systems. Differ- ent techniques will yield slightly different impedance values. In this thesis a simple method based on power losses is used for aggregating a certain number of radials into one equivalent radial.
Consider a WPP with M radials, each with Nm turbines distributed along its radial. Chapter 3. Modelling for harmonic resonance studies 34
Each turbine is assumed to inject the same amount of current. The total current con- tributed from radial m is given by,
N Xm Im = Inm , m = 1, 2, ...M (3.17) nm=1
where Inm is the current from turbine n along radial m counting from the end of the radial towards the substation. The series impedance of an equivalent cable representing radial m can be obtained.
Nm 2 2 X nm Zeqm Im = Znm Im Nm nm=1
Nm 2 X nm ⇒ Zeqm = Znm (3.18) Nm nm=1 where Znm the impedance of the cable section connecting the turbines n and (n + 1) in the m-th radial. The impedance of the M parallel radials can be further reduced into one equivalent radial,
!2 !2 N N Z = 1 Z + 2 Z + ··· MV PM eq,1 PM eq,2 m=1 Nm m=1 Nm !2 N + M Z (3.19) PM eq,M m=1 Nm
The shunt capacitances of all medium voltage radials can be added together, based on the assumption of uniform voltage in the collector system.
N N N X1 X2 Xm CMV = Cn1 + Cn2 + ··· + CnM (3.20) n1=1 n2=1 nm=1
where Cnm is the cable shunt capacitance of the section located in radial m between turbine n and (n + 1) [33]. 4 Test Cases and HRMA Implementation
This chapter is dedicated to HRMA of test cases which have been subject to the similar analysis in published papers. The results obtained thus verify the algorithms which have been implemented in MATLAB for this thesis. In addition, some important considera- tions in implementation as well as relevant contributions from this thesis are highlighted.
4.1 Three Bus Case
A three bus test case with fundamental frequency parameters is shown in Figure 4.1. Reactances and susceptances are scaled in proportion to harmonic order, and resistances are treated as constant in this simplified example. Calculations for both frequency scan and HRMA are carried out using MATLAB. This test case has been subject to the same kind of analysis in several papers [5, 16, 37].
XSYS Bus 1 XL,12 Bus 2 XL,23 Bus 3
0.04+j0.3 0.835+j4 0.835+j4
AC B1 B3 j0.0013 j0.0013
Figure 4.1: Three bus case with component data in per unit. Example system used to illustrate frequency scan and modal impedance scan. All parameters values apply at fundamental frequency.
Considering a frequency spectrum from 1 to 60 pu, there are two resonances in this three bus system. Because the system is so simple, resonances can be identified analytically.
One parallel resonance is associated to the frequency where XB1 is equal in magnitude to XSYS||(XL,12 + XL,23)(XB3 is small and can be ignored). This resonance frequency
35 Chapter 4. Test Cases and HRMA Implementation 36 is given approximately by
s v 1 u 1 fres,pu = = u = 51.58 pu XSYS||(XL,12 + XL,23) · B1 t 0.3·8 0.3+8 · 0.0013
Another parallel resonance is associated to the frequency where XB3 is equal in magni- tude to XL,12 + XL,23 + XSYS.(XB1 ≫ XSYS, XB1 can be ignored.) This resonance frequency is given approximately by
s 1 r 1 fres,pu = = = 9.63 pu (XSYS + XL,12 + XL,23)B3 8.3 · 0.0013
Frequency Scan
The frequency scan, or nodal impedance frequency response, is equivalent to plotting the diagonal elements of the impedance matrix [Y ]−1 over the frequency range of interest. The result from such a calculation in the frequency range of 1 to 60 per unit is shown in Figure 4.2. Both resonance frequencies are identifiable from the frequency scan. The resonance near 51.6 pu is most visible at bus 1, the resonance near 9.6 pu is most visible at bus 3, while both resonances can be seen in the driving point impedance at bus 2. The magnitude of driving point impedance at each bus for the two resonance frequencies are listed in Table 4.1.
Z 11 5000 Z 22 Z 33
4000
3000
2000 Impedance Z [pu]
1000
0 0 10 20 30 40 50 Frequency [pu]
Figure 4.2: Frequency scan of three bus system. The two resonance frequencies are detected in the impedance frequency responses. Chapter 4. Test Cases and HRMA Implementation 37
Table 4.1: Driving point impedance magnitudes at resonance frequencies.
Resonance frequency [pu] 51.61 9.62 Z11 [pu] 5637 9.48 Z22 [pu] 1313 1005 Z33 [pu] 17.5 3733
HRMA
The modal impedance frequency responses are shown in Figure 4.3 for the three bus test case. Some papers refers to the modal variables themselves as modes, in which case the modes are defined at every frequency. In other papers the modal impedance resonance peaks are defined as the modes, a definition which is adapted in this thesis. It is common practise to number the modes based on the magnitude of modal impedance at resonance. The modes near 51.61 pu and 9.62 pu are therefore defined as Mode 1 and Mode 2 respectively. The third modal variable has no resonance in the investigated frequency range.
7000
Mode 1 6000
5000 Mode 2
4000
3000 Impedance Zm [pu] 2000
1000
0 0 10 20 30 40 50 60 Frequency [pu]
Figure 4.3: Modal impedance scan of three bus system. There are two modal reso- nances, or simply modes, in the investigated frequency range. The modal impedances are defined at all frequencies and have continous frequency responses.
Comparing the nodal impedance frequency scan of Figure 4.1 and the modal impedance frequency scan of Figure 4.3, one can see each nodal resonance is linked to the excitation of a particular resonance mode. Table 4.2 lists the participation factors for the three defined buses in each of the two resonance modes. Bus 1 has the highest participation factor in Mode 1, and bus 3 has the highest participation factor in Mode 2. Based on this result, Xu et al. [5] suggested the participation factors can indicate the center of resonance, i.e. where the resonating branches are connected. However, it will be Chapter 4. Test Cases and HRMA Implementation 38 shown later that participation factors can be misleading when it comes to locating the resonating branches.
Table 4.2: Participation factors calculated at frequencies of the modal resonance peaks.
Mode 1 Mode 2 Resonance freq. [pu] 51.61 9.62 Modal impedance peak value Zm [pu] 6951 4743 PF bus 1 0.8103 0.0011 PF bus 2 0.1887 0.2121 PF bus 3 0.0011 0.7868
Previously it was shown how driving point impedance at bus i for any frequency can be expressed as,
Zii = Zm,1PFi1 + Zm,2PFi2 + Zm,3PFi3 + ··· + Zm,nPFin (2.25 revisited) or in terms of weighted participation factors,
Zii = WPFi1 + WPFi2 + WPFi3 + ··· + WPFin (2.27 revisited)
In the considered three bus test case, one modal impedance is much larger than the others at the frequencies of the resonance modes. One term on the right hand side of (2.25) is thus much larger than the other terms, i.e. one weighted participation factor is almost equal to the driving point impedance at the resonance frequencies. For example, the weighted participation factor of bus 1 in Mode 1 is given by
WPF11 = Zm,1PF11 = 6951 · 0.8103 = 5632 pu
It may be confirmed that WPF11 ≈ Z11 by comparing WPF11 with the value of Z11 listed in Table 4.1. This indicates that the resonance observed from bus 1 is linked to excitation of Mode 1. Similarly the weighted participation factor of bus 3 in Mode 2 is given by
WPF32 = Zm,2PF32 = 4743 · 0.7868 = 3732 ≈ Z33 (4.1)
It is worth pointing out that the sum of participation factors with respect a mode is equal to one. This can be confirmed by taking the sum of participation factors in each column of Table 4.2. This is due to the fact that eigenvectors calculated with MATLAB’s eig()-function always have a 2-norm equal to one (see AppendixB).
A good indicator of how involved a network element is in a resonance mode is how much the resonance frequency changes when parameters of the network element are changed. Chapter 4. Test Cases and HRMA Implementation 39
This is quantified by the frequency sensitivity index,
∂f ∂f ∂f h pu i res = res = res · α (2.46 revisited) ∂α norm ∂α ∂α 100 % α
Figures 4.4 and 4.5 show for the two resonance modes the frequency and impedance sensitivity index of the network parameters in bar chart and table format. The indices are calculated as outlined in Chapter2. In the bar diagram the parameters are ranked from left to right in descending order of frequency sensitivity index magnitude. Increasing the value of an inductive reactance or capacitive susceptance will generally reduce associated resonance frequencies, hence the frequency sensitivity index is negative in sign. The sign of the impedance sensitivity index is positive for some parameters and negative for others. Parameters with high magnitude of frequency sensitivity index mostly also have high magnitude of impedance sensitivity index. Sensitivity indices for this three bus case have not been presented in past papers.
For the modal resonance peak at h=51.61 pu frequency, also known as Mode 1, the frequency sensitivity is high for B1 and XSYS, and low for every other parameter.
Similarly for Mode 2, the frequency sensitivity index is high for B3, XL,12 and XL,23, and low for other parameters. The frequency sensitivity index hence effectively identifies the network branches most involved in the resonance modes.
Modal resonance at h=51.6124 5 ] α 0
-5
-10
-15
-20
-25 Frequency sensitivity [pu/100%] Modal impedance sensitivity [%/100%] Freq sensitivity [pu/% change of -30 Bus 1 shunt B Bus 1 shunt X Branch 1-2 X Branch 2-3 X Bus 3 shunt B
Figure 4.4: Frequency sensitivity of resonance mode at h=51.61 pu with respect to capacitive susceptances and inductive reactances of the three bus test system. Network branch Freq. sens. [pu/100%] Impedance sens. [%/100%] Bus 1 shunt B -25.7719 -0.9239 Bus 1 shunt X -24.8060 -2.7704 Branch 1-2 X -0.5006 -0.2712 Branch 2-3 X -0.5000 0.1337 Bus 3 shunt B -0.0361 0.0160
A better understanding of the sensitivity indices can be obtained by considering Fig- ure 4.6, which shows movement of the modal resonance peak near h=9.62 per unit frequency following a 5 % increase and decrease of the shunt connected susceptance at Chapter 4. Test Cases and HRMA Implementation 40
Modal resonance at h=9.6204 1 ] α
0
-1
-2
-3
-4 Frequency sensitivity [pu/100%] Modal impedance sensitivity [%/100%] Freq sensitivity [pu/% change of -5 Bus 3 shunt B Branch 1-2 X Branch 2-3 X Bus 1 shunt X Bus 1 shunt B
Figure 4.5: Frequency sensitivity of resonance mode at h=9.62 pu with respect to capacitive susceptances and inductive reactances of the three bus test system. Network branch Freq. sens. [pu/100%] Impedance sens. [%/100%] Bus 3 shunt B -4.8038 -0.9271 Branch 1-2 X -2.3118 0.7041 Branch 2-3 X -2.3118 0.3113 Bus 1 shunt X -0.1866 0.0572 Bus 1 shunt B -0.0067 0.0003
5400
5200 X: 9.87 X: 9.507 Y: 4992 5000 Y: 4902 X: 9.62 Y: 4743 4800 X: 9.738 X: 9.389 Y: 4584 4600 Y: 4517
4400
No parameter change 4200 5 % increase B Modal impedance Zm [pu] 3 5 % decrease B 4000 3 5 % increase X L,12 5 % decrease X 3800 L,12
9.4 9.5 9.6 9.7 9.8 9.9 10 Frequency [pu]
Figure 4.6: Movement of resonance mode peak near 9.62 pu following change of B3 (bus 3 shunt B), and XL,12 ( branch 1-2 X). bus 3 and reactance of branch 1-2. The normalised sensitivity indices with respect to these parameters can be estimated based on the coordinates given in Figure 4.6. The modal sensitivity indices are approximately given by,
∂Z ∂Z B ∆Z B m = m · 3 ≈ m · 3 ∂B3 norm ∂B3 Zm ∆B3 Zm (4517 − 4992) ZB % = · Z3 = −1.001 [ ] (4.2) 0.1 · ZBZ3 4743 100% Chapter 4. Test Cases and HRMA Implementation 41 and
∂Z ∂Z X ∆Z X m = m · L,12 ≈ m · L,12 ∂XL,12 norm ∂XL,12 Zm ∆XL,12 zm (4902 − 4584) @B % = · = 0.6705 [ ] (4.3) 0.1 · @B 4743 100%
The modal frequency sensitivity indices are approximately
∂fres ∂fres ∆fres = · B3 ≈ · B3 ∂B3 norm ∂B3 ∆B3 9.389 − 9.870 pu = · ZBZ3 = −4.810 [ ] (4.4) 0.1 · ZBZ3 100% and
∂fres ∂fres ∆fres = · XL,12 ≈ · XL,12 ∂XL,12 norm ∂XL,12 ∆XL,12 9.507 − 9.738 H pu = H · XHL,H12 = −2.310 [ ] (4.5) 0.1 · XHL,H12 100%
The numerical estimates of frequency sensitivity index shown in (4.4) and (4.5) are very close to the ones obtained in simulation. The numerical estimates of impedance sensitivity index shown in (4.2) and (4.3) are relatively close, but with some deviation compared to simulation results. The discrepancy between numerical estimate and sim- ulation is likely due to the impedance-parameter relationship not being exactly linear. Based on this kind of simple calculation one can see the usefulness of sensitivity indices for predicting movement of the modal resonance peaks in the impedance-frequency plane.
4.2 IEEE 14 Bus Case
The IEEE 14 bus test system is in this section analysed using HRMA. Despite this system representing a power distribution network, not an offshore WPP, it is chosen for analysis because results from HRMA of this system are available in published papers [6,7]. The details of the IEEE 14 bus test case can be found in [38]. Bus and branch data for the case is included in AppendixA. The frequency range of interest is from fundamental frequency up to 30 pu due to this range being considered in the aforementioned papers.
The following key points are considered in construction of the admittance matrix for frequency scan and HRMA: Chapter 4. Test Cases and HRMA Implementation 42
Figure 4.7: Single line diagram of IEEE 14 bus system [38]. G indicates generator and C indicates synchronous condenser.
• The generators at buses 1 and 2 and the synchronous condenser at bus 6 are all modelled by their subtransient reactances, which is equal to 0.25 pu at fundamental frequency.
• Transformers are modelled by their respective short-circuit reactances.
• The loads are modelled using the CIGRE load model, based on the fundamental frequency load flow included in AppendixA.
• All harmonic filters are of the single tuned type [12].
• The harmonic sources, i.e. the twelve-pulse HVDC converter at bus 3 and SVC at bus 8, are assumed to behave like current sources and are thus ignored in construction of the admittance matrix for frequency scan and HRMA.
• Frequency dependency of parameters is ignored.
• The lines which have specified values of charging susceptance are modelled using the long line corrected π-equivalent, with the correction factors calculated at each discrete frequency. Lines without specified values of charging susceptance are modelled using the short-line approximation.
The driving point impedance frequency responses are plotted in Figure 4.8, and the modal impedance scan is shown in Figure 4.9. The decoupling of the resonances achieved by the modal analysis approach is evident. However, unlike for the three bus case, some modal impedances do exhibit more than one local maxima. Hence a modal variable can not be associated to a single resonance mode. Chapter 4. Test Cases and HRMA Implementation 43
Bus1 9 Bus2 Bus3 8 Bus4 Bus5 7 Bus6 Bus7 6 Bus8 Bus9 5 Bus10 Bus11 4 Bus12 Bus13 Impedance Z [pu] 3 Bus14
2
1
0 0 5 10 15 20 25 30 Frequency [pu]
Figure 4.8: Frequency scan of all buses in IEEE 14 bus test system.
20
15
10
Modal impedance Zm [pu] 5
0 5 10 15 20 25 30 Frequency [pu]
Figure 4.9: Modal impedance scan of IEEE 14 bus system. Certain modal variables have several resoance peaks in the investigated freuqency range. The modal scan is similar to the one obtained in papers [6,7].
The envelope of the modal impedance curves has 9 local maxima in the frequency range up to 30 pu. The participation factors for buses in each if these 9 resonance modes have been calculated and results are included in AppendixA. Table 4.3 sums up the resonance modes in terms of mode frequency and magnitude, the most participating bus and its participation factor.
The weighted participation factor of the most participating bus in each resonance mode Chapter 4. Test Cases and HRMA Implementation 44
Table 4.3: IEEE 14 bus system, magnitude and frequency of envelope local maxima, most participating bus and its participation factor.
Modal resonance Modal impedance Largest Most partici- freq order h [pu] magnitude [pu] PF pating bus # 1.917 4.4553 0.2705 8 2.869 9.3417 0.2632 3 4.212 7.9056 0.6543 8 5.874 4.5109 0.4359 8 8.627 6.8356 0.1376 9 10.738 12.4556 0.2027 1 14.458 13.1867 0.3618 9 25.261 22.5070 0.4138 5 25.943 18.0491 0.3742 1 can be found by multiplying the modal impedance magnitude with the participation fac- tor of that bus, i.e. by multiplying elements in the second and third column in Figure 4.3. Principally the weighted participation factor of the most participating bus approaches the largest driving point impedance at the resonance frequencies. For example, at the resonance mode near 14.458 pu frequency,
WPF39 = 13.1867 · 0.3618 = 4.868 ≈ Z99
The actual value of Z99 can be observed in Figure 4.8. Similar results can be obtained for the other most participating buses listed in Table 4.3. The driving point impedance resonances can hence predominantly be linked to excitation of individual resonance mode.
Figure 4.10: Sensitivity indices of the five most participating branches in modal resonance located at h=4.212 pu frequency.
Modal resonance at h=4.2116 1 ] α
0.5
0
-0.5
-1 Frequency sensitivity [pu/100%] Modal impedance sensitivity [%/100%] Freq sensitivity [pu/% change of -1.5 Bus 8 filter 5th B Bus 8 filter 5th X Branch 7-8 transf. X Bus 8 filter 7th B Branch 4-7 transf. X Network branch Freq. sens. [pu/100%] Impedance sens. [%/100%] Bus 8 filter 5th B -1.4578 -1.2826 Bus 8 filter 5th X -1.0477 -0.9098 Branch 7-8 transf. X -0.3477 0.5649 Bus 8 filter 7th B -0.2598 0.1089 Branch 4-7 transf. X -0.1522 0.3382 Chapter 4. Test Cases and HRMA Implementation 45
The sensitivity indices can also in this case be used to locate the most participating net- work branches in resonance modes of interest, and to predict movement of the resonance mode in the frequency-impedance plane following a parameter change. As an example, the sensitivity indices are shown in Figure 4.10 for the five parameters that have the largest frequency sensitivity index with respect to the resonance mode at 4.212 pu fre- quency. The sensitivity indices show that the 5th harmonic filter located at bus 8 is heavily involved in this resonance mode. In this case the most participating bus is also the location of the network component with highest sensitivity indices.
9 5 % decrease of B No parameter change X: 4.285 5 % increase of B Y: 8.425 8.5
X: 4.21 Y: 7.905 8
X: 4.138 Y: 7.404 7.5 Modal impedance Zm [pu] 7
6.5 3.95 4 4.05 4.1 4.15 4.2 4.25 4.3 4.35 4.4 4.45 Frequency [pu]
Figure 4.11: Movement h=4.212 pu modal resonance with 5 % variation of 5th har- monic filter capacitive susceptance B1 at bus 8.
Figure 4.11 shows how the resonance mode around harmonic order h=4.212 pu changes with a ±5 % parameter change of capacitive susceptance B1 of the 5th harmonic filter at bus 8. A 5 % parameter increase reduces the magnitude and frequency of the modal resonance, while a 5 % decrease of the capacitive susceptance has the opposite effect. The coordinates of the modal impedance peaks are marked in the figure, and allows for numerical verification of sensitivity indices. The impedance sensitivity is approximately
∂Z ∂Z B ∆Z B m = m · ≈ m · ∂B norm ∂B Zm ∆B zm (7.404 − 8.425) @B % = · = −1.292 [ ] 0.1 · @B 7.905 100% Chapter 4. Test Cases and HRMA Implementation 46 and the frequency sensitivity is approximately
∂f ∂f ∆f res = res · B ≈ res · B ∂B norm ∂B ∆B 4.138 − 4.285 pu = · @B = −1.470 [ ] 0.1 · @B 100%
Both of these estimates are very close to the values obtained in simulation which are listed in the table of Figure 4.10.
Lines and cables often play a crucial role in harmonic resonances in power system. Hence the importance of accurate line and cable models for harmonic studies must be stressed. Figure 4.12 shows a comparison of the driving point impedance at bus 3 when the lines are modelled using short line models, nominal π-equivalent models and long-line corrected π-equivalent models. The long line effects are prominent from about the 25th harmonic order and above. A similar figure and associated observations can be found in the documentation of this 14 bus test system [38].
Short line equivalents 4.5 Nominal π-equivalents Long-line corrected π-equivalents 4
3.5
3
2.5
2
Impedance Z [pu] 1.5
1
0.5
0 0 5 10 15 20 25 30 35 40 Frequency [pu]
Figure 4.12: Frequency scan at bus 3 showing effect of various cable models in fre- quency range up to 40th harmonic. Long line effects are prominent from about the 25th harmonic order and higher.
The modal impedance scan obtained using nominal π-equivalent line models is shown in Figure 4.13. Comparing with Figure 4.9, the two resonance modes of highest frequency are different. It can be verified that several lines are heavily involved in these resonances. Long line correction factors hence should not be neglected when analysing resonances where lines or cables are heavily involved. Chapter 4. Test Cases and HRMA Implementation 47
25
20
15
10 Modal impedance Zm [pu] 5
0 5 10 15 20 25 30 Frequency [pu]
Figure 4.13: Modal impedance scan of IEEE 14 bus system using nominal π- equivlaent line models.
4.3 HRMA Implementation
Results obtained from the three and 14 bus test cases are mostly consistent with the what has been presented in past papers. This inspires confidence in the HRMA al- gorithms implemented in MATLAB. This section is used to highlight some important considerations in implementation of HRMA. In the authors opinion some aspects have not been properly presented in past papers, hence the presentation here is considered a contribution of this thesis.
Modal Switching
Modal impedances plotted against frequency generally appear as smooth, continuous curves, exemplified by Figure 4.3 and Figure 4.9. An issue encountered when imple- menting the algorithms for HRMA is that the standard MATLAB eig()-function, which returns the eigenvectors and eigenvalues of a matrix, does not sort the eigenvalues and eigenvectors in a predictable way. As an example consider the Y-bus matrix of a three bus system. The eigenvalues and eigenvectors at discrete frequency f are returned by the eig()-function respectively in the diagonal [Λf ]-matrix, and as columns of [Lf ],
. . . λ1,f 0 0 . . . [Λf ] = 0 λ2,f 0 , [Lf ] = l1,f l2,f l3,f (4.6) . . . 0 0 λ3,f . . . Chapter 4. Test Cases and HRMA Implementation 48
where for example l1,f is the eigenvector of λ1,f , the inverse of which is the modal
impedance Zm,1. At the next discrete frequency, separated from f by a small step ∆f, there will be minor changes in eigenvalues and eigenvectors. However, the random nature in which the eigenvalues are returned by the eig()-function might result in a new order of the eigenvalues, for example,
. . . λ2,f+∆f 0 0 . . . [Λf+∆f ] = 0 λ3,f+∆f 0 , [Lf+∆f ] = l2,f+∆f l3,f+∆f l1,f+∆f . . . 0 0 λ1,f+∆f . . . (4.7) In other words the eigenvalue associated to a particular modal impedance can be re- turned from the eig()-function at a seemingly random position in the diagonal Λ-matrix. Modal switching refers to such swapping of places in the eigenvalue and eigenvector ma- trices while obtaining the frequency response. If no countermeasures are taken, the resulting modal impedance scan might look like the one in Figure 4.14. Such plots are presented in several papers on the topic of HRMA [21, 24, 25].
20
15
10
Modal impedance Zm [pu] 5
0 5 10 15 20 25 30 Frequency [pu]
Figure 4.14: Modal switching observed when analysing the 14-bus test system without countermeasures. The eigenvalues swap places in the diagonal eigenvalue matrix Λ as the frequency is sweept.
There are several ways to deal with the issue of modal switching. A straightforward way to approach the problem is to use a combination of the eig()-function and a correlation technique to reorder the eigenvalues and eigenvectors whenever a switch occurs. Corre- lation technique is one of three methods covered in a paper by Zlatuni´cet al. [39]. From basic matrix algebra it is known that eigenvectors associated with distinct eigenvalues are orthogonal to each other, i.e. they have a vector-product equal to zero. Moreover, eigenvectors returned by the eig()-function are normalised so that the 2-norm of each Chapter 4. Test Cases and HRMA Implementation 49 eigenvector is equal to one. Hence multiplying an eigenvector with itself gives a product of one, and multiplying eigenvectors associated with different eigenvalues gives a product of zero. The correlation technique can be summarized as follows:
• Obtain the eigenvector-matrix Lf−∆f for the previous discrete frequency and
eigenvector-matrix Lf for the current frequency which is increased by ∆f from the previous step.
• Obtain the matrix product of the transpose eigenvector-matrix from current step T with the eigenvector matrix from previous step, Lf · Lf−∆f .
• In each column of the aforementioned matrix-product, one element will be close to unity, while the other elements will be close to zero. The row and column number of the unity-elements define the column numbers of associated eigenvectors at the current and previous frequency respectively.
• Sort the eigenvectors and eigenvalues at the current frequency according to the order of the previous frequency.
With this correlation technique, the result at the initial frequency defines the order of the eigenvalues and eigenvectors. Mode switching is inevitable, but is tracked and corrected as one moves from one frequency to the next. Because eigenvalues and eigenvectors change a little as on moves from one discrete frequency to the next, a relatively small step between discrete frequencies is required for the method to work reliably. In the authors experience with the test cases, discrete steps of 0.01 pu frequency and smaller gives reliable correction of modal switching.
An entirely different approach is to ignore the issue of modal switching and only consider the envelope of the modal impedance curves. At a modal resonance frequency, one eigenvalue is usually much smaller than the others, hence there is only one large modal impedance, and this impedance is visible in the envelope of modal impedance scan. However, if several resonances appear at frequencies close to each other, resonance peaks may not appear in the envelope. The modal envelope approach is referred to as critcal mode scan by Xu et al. [5].
Complex vs. Real Number Admittance Matrix
An admittance matrix is usually formulated as a matrix of complex numbers, but can however also be formulated as a matrix of real numbers. By splitting the real and Chapter 4. Test Cases and HRMA Implementation 50 imaginary parts of phasors, the real number Y-matrix for a 2-node network is obtained.
−1 V1x G11 G12 −B11 −B12 I1x V G G −B −B I 2x 21 22 21 22 2x = (4.8) V1y B11 B12 G11 G12 I1y V2y B11 B12 G21 G22 I2y where subscripts x and y denote the real and imaginary parts of complex phasors. The real number Y-matrix has twice the rank and therefore twice the number of eigenvalues with respect to the complex number Y-matrix. It may be verified that half of the eigenvalues of the real Y-matrix are identical to those of the complex Y-matrix, and the other half are complex conjugate of the first half. Consequently, both the real and complex admittance matrices are suitable for HRMA and will yield the same results [5– 7, 20].
From (4.8) one can see that the conversion from one form to the other is rather straight- forward. Some power analysis software use the real Y -matrix formulation, hence one can envision a real matrix approach to be preferable [5]. An argument can also be made in favour of the real matrix representation due to the eigendecomposition being more efficient for real matrices [20]. However, the complex matrix has been used in this the- sis as it allows for avoiding the complex conjugate set of eigenvalues. In addition, the author perceives the implementation of HRMA sensitivity indices as easier when using complex matrix formulation.
Real Y-matrix Envelope Using Power Method
If only the modal impedance envelope is of interest, the power method can be used. This method calculates only the smallest eigenvalue, and is less computer intensive than algortims which calculate every eigenvalue. Xu et al. [5] suggested using the power method for calculation of the modal envelope, and found the real admittance matrix to be preferable. They also found some discrepancy in critical mode scan envelope obtained with the two different matrix formulations [20].
Calculation of the modal envelope using the real Y-bus matrix and the power method was carried out to investigate if there really is a difference in the envelope obtained with the two different Y-bus matrix formulations. Figure 4.15 shows the modal envelope of the 14 bus test system calculated using the real Y-matrix. The critical eigenvalue was calculated using the power method algorithm, and the number of iterations required for an accuracy of 0.005 is shown on the right y-axis. The power method is known to require a high number of iterations when two eigenvalues are of equal magnitude. The Chapter 4. Test Cases and HRMA Implementation 51
10000
20
5000 Modal impedance Zm [pu] Number of iterations for power method
0 0 5 10 15 20 25 30
Figure 4.15: Modal impedance envelope and number of iterations using real admit- tance matrix formulation and power method with accuracy of 0.005. Good agreement with complex admittance formulation is obtained. Notice the high number of iterations required when two eigenvalues are of similar magnitude. agreement with the envelope of modal impedance scan of Figure 4.9 is excellent, with none of the discrepancy seen in aforementioned papers. Hence the discrepancy which was identified in past papers can most likely be attributed to a poor implementation of the power method.
Sensitivity Indices and Non-RLC Branch Network Elements
The information provided by the participation factors as well as modal impedance sen- sitivity index and frequency sensitivity index was demonstrated for the test cases of this chapter. Among these, participation factors are by far the least burdensome to calculate in terms of computational effort required. Impedance and frequency sensitivity indices require far more computational time and/or power. They rely on numerical calculation of several terms, and experience with the presented test cases suggest discrete steps of frequency as low as 1 × 10−4 pu are required for decent accuracy.
Calculation of sensitivity indices require the explicit calculation of terms such as ∂Gbr/∂α∂h, where α can be the fundamental frequency parameter R1, X1 or B1. In order to obtain a general algoritm it was assumed that all relevant network branches of interest are fully representable by an RLC-branch. This is not the case for cables or lines when long line correction factors are taken into account. The hyperbolic correction factors of the long line corrected model makes it difficult to derive straightforward expressions for sensitivity indices. Hence it is arguably easier to simply calculate the sensitivity indices numerically. This can be done by recalculating the modal impedance resonance Chapter 4. Test Cases and HRMA Implementation 52 magnitude and frequency when α is subject to a small change of value, i.e. ±∆α. Such an approach is taken in this thesis whenever long line corrected models are used. 5 Offshore WPP Resonance Analysis
In this chapter HRMA is applied for resonance analysis of a WPP model based on the real 400 MW Anholt offshore WPP.
HRMA has in the past been applied to offshore WPPs [22, 24]. In those papers a num- ber of simplifying assumptions are used in modelling of the WPPs, such as uniform and heavily aggregated radials in the array collector system. Moreover they constrain anal- ysis to below the 25th harmonic frequency, a choice likely motivated by the limitations of the nominal π-equivalent cable models. Modal switching is not corrected, resulting in some confusion in terminology and results. Further investigation of the capabilities of the HRMA method when it comes to resonance analysis of offshore WPPs is therefore warranted.
The sensitivity of HRMA results with respect to the following aspects and simplifications in modelling of the WPP will be investigated:
• WPP radial collector system aggregation
• Type of cable models, nominal or long-line corrected π-equivalents
• Uniform collector radials
• Frequency dependent resistance/skin effect
• WTG modelling, emission filters and converter current controller
5.1 Anholt Offshore WPP Layout
The 400 MW Anholt offshore WPP is comprised of 111 turbines, each rated at 3.6 MW and equipped with a full-scale frequency converter. An overview of the WPP layout and export system is shown in Figure 5.1. As indicated in the figure, there are three 140 MVA transformers located at the offshore substation, and the points of common coupling (PCC) are defined on the 33 kV side of these transformers. Downstream of the
53 Chapter 5. Offshore WPP Resonance Analysis 54
PCCs are the collector array cables and wind turbines. This part of the system was installed and is operated DONG Energy Wind Power. The Danish TSO, Energinet.dk, is responsible for the part of the system which is upstream of the PCC, i.e. the export system.
WTs 33/220 kV WTs 140 MVA
WTs PCC #1 2x60 MVAr WTs
220/400 kV WTs 33/220 kV WTs 140 MVA Submarine cable Land cable 220/400 kV WTs PCC #2 120 MVAr
WTs 2x60 MVAr Shunt compensation WTs 33/220 kV WTs 140 MVA Onshore substation
WTs PCC #3
WTs
WPP and offshore substation
Total array Total capacitance Connected to WTGs Arrays cable length of array cables PCC #1 37 4 50km 14uF PCC #2 37 4 48km 13uF PCC #3 37 4 54km 15uF
Figure 5.1: Single line diagram of Anholt WPP and export system. Table shows key data about radial arrays downstream of each point of common coupling [32].
Each 140 MVA transformer connects 37 WTGs across 4 radials to the 220 kV export system. Hence there are in total 12 radials, each one collecting the power from either 9 or 10 turbines. An overview of the array collector system is given in AppendixC. The array layout is non-uniform, and each radial uses a combination of 150 mm2, 240 mm2 and 500 mm2 copper cables. There is also a difference in total length and capacitance of the array cables connected to each PCC, as indicated in the table of Figure 5.1.
The high voltage (220 kV) export system includes one 24.5 km aluminium 3x1600 mm2 submarine cable, and one 58 km aluminium 3x2000 mm2 underground cable. There is Chapter 5. Offshore WPP Resonance Analysis 55 reactive compensation of 120 MVAr placed in between the cables. The onshore substa- tion includes 2x450 MVA transformers connecting the underground cables to the central grid, as well as another 4x60 MVAr of reactive compensation.
Beyond the aforementioned data, little is public knowledge when it comes to parameter values of the Anholt WPP. Hence it is not an objective to achieve a model with the exact parameter values of the real system. Cable data is estimated based on ABB’s user’s guide for submarine and land cable systems [40, 41], and transformer data is estimated roughly based on rule of thumb.
Wind Power Plant Models
Three different WPP models with varying degree of aggregation are implemented for HRMA. All models are based on the simplifying assumption that the four radials con- nected to each PCC are symmetrical. Those four radials are further reduced into one equivalent radial using the aggregation technique of Section 3.6. Assuming there are 9 turbines in each radial, this aggregation results in a WPP model with 27 turbine equivalents and 33 buses as shown in Figure 5.2. The total length and capacitance of radials connected to each PCC is given in Figure 5.1. In obtaining the 33 bus model the total length of the four aggregated radials at each PCC is set equal to the listed value. Cross-section and share of total length for each cable section is selected to achieve good fitting with the listed capacitance values.
The 33 bus WPP model has a relatively high number of buses and branches, and this might make it unnecessarily complicated for getting an overview of resonances in the system. Two models with higher degree of radial collector cable aggregation are therefore also defined. Those models are further reductions of the 33 bus model obtained using the aggregation technique of Section 3.6. A 15 bus model, shown in Figure 5.3, represents all collector cable sections with the same cross-section as a single cable and with 3 equivalent WTGs per radial. The 9 bus model, shown in Figure 5.4, represents each radial with one cable and one WTG equivalent.
Wind turbines, including main reactors, shunt harmonic filter and transformers, are initially modelled as harmonic current source, hence these components do not appear in the Y-bus matrix. It can also be regarded as a situation where all turbines are disconnected and WTG main circuits are de-energised, i.e. transformers, shunt harmonic filters and main reactors are disconnected.
The modal envelope and participation factors can be obtained for the most compre- hensive model without much computational burden. However, calculation of the modal Chapter 5. Offshore WPP Resonance Analysis 56
Bus 31 Bus 10 Bus 7 Bus 4/PCC#1
Bus 5/ Bus 3 Bus 32 Bus 11 Bus 8 PCC #2
Bus 6/ Bus 33 Bus 12 Bus 9 PCC #3 Submarine cable Midpoint reactor Cable Substation sections 3-8 transformers Bus 2 and WTGs Equivalent Cable Equivalent Cable Equivalent Cable WTGs section 9 WTGs section 2 WTGs section 1 Underground cable
Collector cable sections:
Section 9 Section 8 Section 7 Section 6 Section 5 Section 4 Section 3 Section 2 Section 1 Onshore Bus 1 150mm2 150mm2 150mm2 150mm2 150mm2 240mm2 240mm2 500mm2 500mm2 substation
Figure 5.2: Single line diagram of Anholt offshore WPP model with simplified array representation. 33 buses are defined as given in the figure.
sensitivity indices becomes burdensome with the large 33 bus model. It is generally observed that the most critical resonances appear in the simplified 9 and 15 bus models. Figure 5.5 shows a comparison of the modal impedance envelope obtained with the three WPP models with long line corrected cables and skin effect included. Because, the main resonances are reflected in all three models, the 9 bus model is used to gain further insight about these resonances. Chapter 5. Offshore WPP Resonance Analysis 57
Bus 13 Bus 10 Bus 7 Bus 4/PCC#1
Bus 5/ Bus 3 Bus 14 Bus 11 Bus 8 PCC #2
Bus 6/ Bus 15 Bus 12 Bus 9 PCC #3 Submarine cable Midpoint reactor Substation transformers Bus 2 Equivalent Equivalent Equivalent Equivalent Equivalent Equivalent 150mm2 240mm2 500mm2 WTGs WTGs WTGs cables cables cables Underground cable
Onshore Bus 1 substation
Figure 5.3: Single line diagram of Anholt offshore WPP model with simplified array representation. 15 buses are defined as given in the figure.
Bus 7 Bus 4/PCC#1
Bus 5/ Bus 3 Bus 2 Bus 1 Bus 8 PCC #2
Bus 6/ Bus 9 PCC #3 Submarine Midpoint Underground Onshore cable reactor cable substation
Substation transformers Equivalent WTG Equivalent harmonic subgroup array sources cable
Figure 5.4: Single line diagram of Anholt offshore WPP model with simplified array representation. 9 buses are defined as given in the figure. Chapter 5. Offshore WPP Resonance Analysis 58
400 9 bus model 15 bus model 350 33 bus model
300
250
200
150
Modal impedance Zm [pu] 100
50
0 5 10 15 20 25 30 35 40 45 50 Frequency [pu]
Figure 5.5: Modal envelope of 9, 15 and 33 bus models with skin effect included. The same resonances are visible in the evelope of all WPP models. The 9 bus model is therefore considered adequate for investigating these resonances. Chapter 5. Offshore WPP Resonance Analysis 59
5.2 9 Bus Model HRMA
Nominal π-equivalent cable models
HRMA is now carried out for the 9 bus WPP model constructed with nominal π-cable models. The simple structure of this WPP model yields predictable results and al- lows for calculation of modal sensitivity indices as outlined in Chapter2. The modal impedance frequency response is shown in Figure 5.6. There are six resonance modes in the investigated frequency range, with no modal variable involved in more than one resonance mode. Bus participation factors associated to the resonance modes are shown in Table 5.1.
300
250
200
150
100 Modal impedance Zm [pu]
50
0 5 10 15 20 25 30 35 40 45 50 Frequency [pu]
Figure 5.6: Modal impedance scan of 9 bus WPP model with nominal cable mod- els. Modes are numbered in descending order of modal impedance at their respective resonance frequencies.
Table 5.1: Participation factors in resonance modes of 9 bus WPP model with nominal cable models. The largest participation factor in each resonance mode is underlined.
fres [pu] 4.340 12.920 15.623 16.246 16.841 26.597 Zm [pu] 37.320 48.229 183.986 260.900 323.853 17.150 Bus 1 0.0256 0.1023 0.0024 0.0005 0.0001 0.0003 Bus 2 0.0977 0.0013 0.0033 0.0011 0.0003 0.0151 Bus 3 0.1104 0.0199 0.0002 0.0002 0.0001 0.2879 Bus 4 0.1263 0.1255 0.0153 0.3499 0.0488 0.0670 Bus 5 0.1252 0.1041 0.0061 0.0263 0.3744 0.0909 Bus 6 0.1277 0.1670 0.3976 0.0489 0.0058 0.0450 Bus 7 0.1289 0.1506 0.0201 0.4697 0.0671 0.1640 Bus 8 0.1275 0.1223 0.0077 0.0340 0.4947 0.1957 Bus 9 0.1307 0.2070 0.5474 0.0693 0.0085 0.1342 Chapter 5. Offshore WPP Resonance Analysis 60
The participation factors indicate at which buses the resonance modes are most ob- servable in the driving point impedance. For example, the resonance mode near the 16.841 pu frequency is most observable at bus 5 and 8. Similarly, the resonance mode near 16.25 pu is most observable at bus 4 and bus 7, and the the resonance mode near 15.62 pu is most observable at bus 6 and bus 9. The modal sensitivity indices of the five parameters with highest frequency sensitivity with respect to these three resonance modes are shown in Figure 5.7, Figure 5.8, and Figure 5.9 respectively. Frequency sen- sitivity indices reveal that these resonance are due to the collector cables resonating with the offshore substation transformers. For example, the modal resonance frequency near 16.84 pu is most sensitive to shunt capacitance of cable section 5-8 and reactance of transformer 3-5.
In addition to the three resonance modes due to offshore transformers and collector cables, there is one resonance mode at 4.34 pu, another at 12.92 pu and a third at 26.60 pu frequency. The resonance mode at 4.357 pu frequency stands out due every bus except for bus 1 having a participation factor of comparable magnitude. This indicates that, with the exception of bus 1, observability of this mode is similar in the driving point impedance at all buses. The driving point impedance at every bus near this frequency is shown in Figure 5.10, and the impedance magnitudes are clearly proportional to the bus participation factors.
The sensitivity indices of the five parameters with highest frequency sensitivity with respect to the 4.34 pu resonance mode are shown in Figure 5.11. They reveal this mode is due to interaction between the grid reactance (i.e. onshore substation transformer), underground and submarine export cables. Notice the participation factors for this res- onance mode are smallest close to where the resonating branches are connected. This illustrates the important distinction between observability, which is quantified by par- ticipation factors, and branch involvement in the resonance mode, which is quantified by the frequency sensitivity index.
Similar analysis can be carried out for the other resonance modes. It can be verified that the resonance mode at 12.92 pu is due to the underground export cable resonating with the onshore substation transformer. The resonance mode at 26.60 pu frequency is due to interaction between the shunt capacitance and series reactance of the submarine export cable. Sensitivity indices for these resonance modes are included in AppendixC. Chapter 5. Offshore WPP Resonance Analysis 61
Modal resonance at h=16.841 1 ] α 0
-1
-2
-3
-4
-5 Frequency sensitivity [pu/100%] -6 Modal impedance sensitivity [%/100%] Freq sensitivity [pu/% change of -7 Long line 5-8 B PCC2 transf. 3-5 X Long line 4-7 B PCC1 transf. 3-4 X Long line 5-8 X
Figure 5.7: Sensitivty indices of the five parameters with highest frequency sensitivty index with respect to the modal resoance at 16.84 pu frequency. The collector cable of section 5-8 and the transformer of section 3-5 are the network components most involved in the resonance. Modal resonance at h=16.2458 2 ] α
0
-2
-4 Frequency sensitivity [pu/100%] Modal impedance sensitivity [%/100%] Freq sensitivity [pu/% change of -6 Long line 4-7 B PCC1 transf. 3-4 X Long line 6-9 B PCC3 transf. 3-6 X Long line 5-8 B
Figure 5.8: Sensitivty indices of the five parameters with highest frequency sensitivty index with respect to the modal resoance at 16.25 pu frequency. The collector cable of section 4-7 and the transformer of section 3-4 are the network components most involved in the resonance. Modal resonance at h=15.623 4 ] α
2
0
-2
-4 Frequency sensitivity [pu/100%] Modal impedance sensitivity [%/100%] Freq sensitivity [pu/% change of -6 Long line 6-9 B PCC3 transf. 3-6 X Long line 1-2 B Long line 1-2 X Long line 2-3 X
Figure 5.9: Sensitivty indices of the five parameters with highest frequency sensitivty index with respect to the modal resoance at 15.62 pu frequency. The collector cable of section 6-9 and the transformer of section 3-6 are the network components most involved in the resonance. Chapter 5. Offshore WPP Resonance Analysis 62
5 Bus1 4.5 Bus2 Bus3 4 Bus4 Bus5 3.5 Bus6 Bus7 3 Bus8 2.5 Bus9
2 Impedance Z [pu] 1.5
1
0.5
0 4.2 4.25 4.3 4.35 4.4 4.45 4.5 4.55 4.6 4.65 4.7 Frequency [pu]
Figure 5.10: Driving point impedance around 4.36 pu resonance frequency for every defined bus. The driving point impedance amplitudes are roughly proprtional to the participation factors in the resonance mode, listed in the first column of Table 5.1.
Modal resonance at h=4.34 0.5 ] α
0
-0.5
-1 Frequency sensitivity [pu/100%] Modal impedance sensitivity [%/100%] Freq sensitivity [pu/% change of -1.5 Long line 1-2 B Grid reactance. X Long line 1-2 X Long line 2-3 B Long line 6-9 B
Figure 5.11: Sensitivty indices of the five parameters with highest frequency sensi- tivty index with respect to the modal resoance at4.36 pu frequency. The export cables resonate with the reactance of the onshore substation transformers. Chapter 5. Offshore WPP Resonance Analysis 63
Radial symmetry
The driving point impedances obtained at buses 7, 8 and 9 are shown in Figure 5.12 in the vicinity of the cable-offshore transformer resonance modes. At each of the three buses, one mode clearly dominates the impedance response.
160 Bus7 140 Bus8 Bus9
120
100
80
60 Impedance Z [pu]
40
20
0 12 13 14 15 16 17 18 19 Frequency [pu]
Figure 5.12: Driving point impedance obtained at end of collector radials 9 bus WPP model with asymmetric radials and nominal cable modes.
Let’s now investigate what happens to the three resonance modes in the 15 pu to 17 pu frequency range when the three radials of the WPP modal are made symmetrical. The driving point impedance and modal impedance frequency responses are shown in Fig- ure 5.13 respectively on the left and right hand side. The driving point impedance frequency response at buses 7, 8 and 9 become identical. Two of the modal variables now have the same frequency response, resulting in two resonance modes in the same location near 16.44 pu frequency. The third modal variable has a different frequency response, with a resonance mode near the 15.93 pu frequency.
Care must be taken in HRMA when two modal variables have identical frequency re- sponses, as is the case for two modal variables in Figure 5.13. Two of the resonance modes are seemingly identical, but the participation of corresponding buses in different radials, for example buses 7, 8 and 9, are different. as given in Table 5.2. Bus 9 par- ticipates most in Mode 2, bus 8 participates most in Mode 1, while bus 7 participates equally in both resonance modes. Notice the sum of participation factors in Mode 1 and Mode 2 for each bus is equal, hence the driving point impedance at each bus is also equal. Because two modes are identical, analysing only the modal impedance envelope would obviously be insufficient in this case. Chapter 5. Offshore WPP Resonance Analysis 64
Mode 1 120 Bus 7 Bus 8 300 Mode 2 Bus 9 100 250
80 200
60 150 Mode 3 Impedance Z [pu]
40 Impedance Zm [pu] 100
20 50
0 0 12 14 16 18 20 22 12 14 16 18 20 22 Frequency [pu] Frequency [pu]
Figure 5.13: Driving point impedance at buses 7, 8, and 9 (l.h. figure), and modal scan of three most critical modes (r.h. figure) with symmetric radials. Driving point impedance is identical for the three buses at the end of the radials. The frequency response of two modal variables are identical, while the third modal variable has a differenct frequency response.
Table 5.2: Particpation factors of buses 7, 8, and 9 in two most critical modal re- soances, i.e. Mode 1 and and Mode 2
Mode 1 Mode 2 Resonance freq. [pu] 16.44 16.44 Modal impedance peak value Zm [pu] 344.0 344.0 PF bus 7 0.1917 0.1916 PF bus 8 0.3576 0.0257 PF bus 9 0.0257 0.3577
Long line corrected cable models
In the WPP models presented this far, there are no capacitances besides those associated to cables. Any detected resonance must therefore involve one or more cable capacitances. In the analysis of the 14 bus test case it was suggested that long line corrected cable models should be used when analysing resonances where cable capacitances are heavily involved. HMRA is in this section carried out with such cable models.
The modal impedance scan obtained for the 9 bus WPP model with long line cor- rected cable models is shown in Figure 5.14. The overall characteristics of the modal impedances are rather similar to the case with nominal cable models, though there are some differences. Bus participation factors for the 7 detected resonance modes are shown in Table 5.3.
Investigation of the resonance modes can be carried out in a similar fashion to the case with nominal cable models. Participation factors quantify the impact of resonance modes in the impedance characteristic at different buses. Sensitivity indices reveal which Chapter 5. Offshore WPP Resonance Analysis 65
300
250
200
150
100 Modal impedance Zm [pu]
50
0 5 10 15 20 25 30 35 40 45 50 Frequency [pu]
Figure 5.14: Modal scan of 9 bus Anholt WPP. There are seven resonance modes, each in a different modal variable.
Table 5.3: Participation factors of modal resonances. The largest PF of each mode is marked with solid underline.
fres [pu] 4.356 13.460 15.864 16.569 18.296 28.580 41.298 Zm [pu] 36.265 101.126 274.587 304.823 132.144 9.313 8.440 Bus 1 0.0281 0.0130 0.0003 0.0002 0.0145 0.2640 0.3219 Bus 2 0.0973 0.0005 0.0001 0.0001 0.0134 0.0101 0.2720 Bus 3 0.1100 0.0151 0.0004 0.0003 0.0099 0.3195 0.3390 Bus 4 0.1259 0.1324 0.1391 0.1621 0.1073 0.0355 0.0010 Bus 5 0.1249 0.1032 0.0275 0.2514 0.2354 0.0485 0.0020 Bus 6 0.1274 0.1980 0.2545 0.0131 0.0503 0.0235 0.0003 Bus 7 0.1286 0.1625 0.1855 0.2222 0.1583 0.0995 0.0212 Bus 8 0.1272 0.1235 0.0354 0.3316 0.3312 0.1176 0.0231 Bus 9 0.1306 0.2518 0.3572 0.0190 0.0797 0.0818 0.0195 parameters the modes are most sensitive to, and hence which circuit elements are most involved in the resonance modes.
Though sensitivity indices are not included, it may be verified the resonance modes at 4.36 pu and 13.46 pu frequency are of similar nature to the resonance modes at 4.34 pu and 12.92 pu frequency for the case with nominal cable models. The resonance modes at 28.58 pu and 41.30 pu are due to export cables, and are also most observable at buses 1 and 3, close to the resonating branches.
The three most critical resonance modes, i.e. those of highest amplitude in the modal domain, are located in close proximity between 15 pu and 19 pu frequency. These res- onances are mainly due to array collector cables resonating with offshore substation transformers. However, the characteristics of these modes are somewhat different to the case with nominal cable models. Firstly, two of the modes are most observable at bus 8, Chapter 5. Offshore WPP Resonance Analysis 66
100 Bus7 90 Bus8 Bus9 80
70
60
50
40 Impedance Z [pu] 30
20
10
0 11 12 13 14 15 16 17 18 19 20 21 Frequency [pu]
Figure 5.15: Driving point impedance at buses 7, 8 and 9 near resonance frequecies due to offshore transformers and collector cables. The three most critical modes are observable at more than one of these buses. and one mode is most observable at bus 9. This is evident from the participation factors as well as the driving point impedances shown in Figure 5.15. Secondly, the modes are more collaborative in the sense that frequency sensitivity is of comparable magnitude for more parameters than in the case with nominal cable models. This can be observed in the frequency sensitivities for these modes which are shown in Figure 5.16. The export cables are also rather involved in the resonance at 18.31 pu. Chapter 5. Offshore WPP Resonance Analysis 67
Modal resonance at h=15.874 2 ] α 1
0
-1
-2
-3
-4 Frequency sensitivity [pu/100%] Modal impedance sensitivity [%/100%] Freq sensitivity [pu/% change of -5 Long line 6-9 B PCC3 transf. 3-6 X Long line 4-7 B PCC1 transf. 3-4 X Long line 6-9 X
Modal resonance at h=16.576 1 ] α
0
-1
-2
-3
-4 Frequency sensitivity [pu/100%] Modal impedance sensitivity [%/100%] Freq sensitivity [pu/% change of -5 Long line 5-8 B PCC2 transf. 3-5 X Long line 4-7 B PCC1 transf. 3-4 X Long line 5-8 X
Modal resonance at h=18.307 1 ] α
0
-1
-2
-3 Frequency sensitivity [pu/100%] Modal impedance sensitivity [%/100%] Freq sensitivity [pu/% change of -4 Long line 1-2 X Long line 1-2 B PCC2 transf. 3-5 X Long line 5-8 B Long line 2-3 B
Figure 5.16: Sensitivty indices of network parameters with highest frequency sensi- tivity index for 15.86 pu, 16.57 pu, and 18.31 pu modes.
Skin effect
Skin effect forces the flow of current towards the perimeter of conductors, hence de- creasing effective cross-section and increasing resistance. Up to this point, skin effect has been neglected. The impact of taking skin effect into account via the following empirical formula will now be investigated.
√ R = R1(0.187 + 0.532 h, h > 2.35) (3.3 revisited)
Figure 5.17 shows the modal impedance envelopes calculated with and without skin effect taken into account. The increased resistance due to skin causes damping of the Chapter 5. Offshore WPP Resonance Analysis 68 resonances peaks, but the mode frequencies remain largely unaltered. The two modes of highest frequency are completely damped when including skin effect.
300 Without skin effect With skin effect
250
200
150
100 Modal impedance Zm [pu]
50
0 5 10 15 20 25 30 35 40 45 50 Frequency [pu]
Figure 5.17: Modal impedance envelopes of the 9 bus WPP model with long line corrected cable models, with and without inclusion of skin effect. Skin effect dampens resonance peaks, but has little effect on mode frequencies.
The participation factors calculated with skin effect included are shown in Table 5.4. Because participation factors are a relative measure of mode observability at defined buses, there are only minor changes in the participation factors compared to the case without skin effect. Moreover, due to the normalisation of sensitivity indices, these are also virtually unchanged compared to the case without skin effect.
Table 5.4: Participation factors of modal resonances with skin effect included. The largest participation factor of each mode is marked with solid underline. Only minor changes is resonance frequencies and participation factors observed due to inclusion of skin effect.
fres [pu] 4.349 13.442 15.861 16.566 18.319 Zm [pu] 16.016 26.357 65.896 71.649 29.925 Bus 1 0.0279 0.0128 0.0003 0.0002 0.0138 Bus 2 0.0966 0.0005 0.0001 0.0001 0.0128 Bus 3 0.1097 0.0152 0.0004 0.0003 0.0095 Bus 4 0.1260 0.1324 0.1391 0.1615 0.1083 Bus 5 0.1249 0.1034 0.0275 0.2518 0.2352 Bus 6 0.1274 0.1971 0.2543 0.0131 0.0506 Bus 7 0.1290 0.1630 0.1857 0.2216 0.1596 Bus 8 0.1276 0.1242 0.0354 0.3324 0.3304 Bus 9 0.1310 0.2515 0.3571 0.0190 0.0799 Chapter 5. Offshore WPP Resonance Analysis 69
5.3 9 Bus Model with WTG Terminal Filters
The sensitivity of HRMA results with respect to WTG harmonic models is investigated in this section. As mentioned in Chapter3, harmonic sources are often modelled as harmonic current source in frequency domain harmonic studies. It was previously as- sumed the entire wind turbine circuit, i.e. main reactor, shunt filter and transformer, can be modelled as harmonic current source. Now the wind turbine circuit components are modelled separately, as shown in Figure 5.18. A converter in operation is modelled by short circuiting its terminals. This corresponds to a converter with ideal harmonic rejection, i.e. ideal voltage source behaviour.
Ltr Lmr
Converter connection Csf
Rsf
Figure 5.18: WTG main circuit model with main reactor, step-up transformer and shunt harmonic filter modelled separately. Ideal harmonic rejection, i.e. ideal voltage source behavior of converter when WTG is connected.
WTG Emission Filter
A set of parameters for the LCL filter of a grid-tied VSC can be determined based on knowledge of the converter topology and ratings as well as step-up transformer short- circuit inductance. Suggested parameters in per unit expressed on a power base equal to Anholt per WTG active power rating of 3.6 MW are given by,
Lmr = Ltr = 0.064 pu; Csf = 0.0892 · Cbase = 11.21 pu; Rsf = 0.220 pu
The calculations leading to selection of these parameters are based on well established principles. Further details are included in AppendixD. Chapter 5. Offshore WPP Resonance Analysis 70
HRMA results
The modal impedance envelope is shown in Figure 5.19 for all three WPP models with WTG main circuits energised and all converters connected. Skin effect is also modelled. From the figure one can observe that the shape of the envelope curve is preserved in the aggregated 15 and 9 bus WPP models. The 9 bus model is therefore chosen for further analysis.
45 9 bus model 40 15 bus model 33 bus model 35
30
25
20
15 Modal impedance Zm [pu] 10
5
0 5 10 15 20 25 30 35 40 45 50 Frequency [pu]
Figure 5.19: Modal impedance envelope obtained with WTG main circuits included and all converters connected. The envelope shape is similar for all three WPP models, hence it is assumed the 9 bus model is sufficient.
9
8
7
6
5
4
3
Modal impedance Zm [pu] 2
1
0 5 10 15 20 25 30 35 40 45 50 Frequency [pu]
Figure 5.20: Modal impedance scan obtained with 9 WPP model, skin effect included and all WTGs connected. There are 10 resoance modes, many of which occur at fre- quencies close to each other. The resistor in the capacitive high-pass branch introduces significant damping of the resonance modes. Chapter 5. Offshore WPP Resonance Analysis 71
The modal impedance scan for the 9 bus WPP model with skin effect included and all converters connected is shown in Figure 5.20. Ten resonance modes can be identified in the investigated frequency range. Some of the resonance modes are located close to each other, and several resonance modes do not appear in the modal impedance envelope. Considering only the modal envelope hence become insufficient in this case. Participation factors for each resonance mode are listed in Table 5.5, indicating the observability of each resonance mode in driving point impedances.
Table 5.5: Participation factors of resonance modes obtained with 9 bus WPP model and all wind turbines connected. The largest PF of each mode is marked with solid underline.
fres [pu] 5.155 13.375 13.870 13.980 26.435 27.760 28.640 30.050 41.310 Zm [pu] 7.484 4.554 3.586 3.411 6.222 7.982 9.154 8.639 1.281 Bus 1 0.0740 0.0133 0.0002 0.0001 0.0103 0.0029 0.0019 0.0091 0.3209 Bus 2 0.2200 0.0005 0.0000 0.0000 0.0032 0.0003 0.0001 0.0001 0.2706 Bus 3 0.2129 0.0155 0.0002 0.0001 0.0114 0.0037 0.0024 0.0089 0.3388 Bus 4 0.0963 0.1374 0.1729 0.1266 0.0879 0.2626 0.2276 0.0747 0.0085 Bus 5 0.0950 0.1084 0.0311 0.2966 0.0443 0.0446 0.3090 0.4362 0.0101 Bus 6 0.0980 0.2003 0.2271 0.0117 0.4360 0.2530 0.0179 0.0212 0.0065 Bus 7 0.0680 0.1602 0.2236 0.1675 0.0613 0.1988 0.1845 0.0654 0.0150 Bus 8 0.0683 0.1239 0.0391 0.3811 0.0302 0.0327 0.2412 0.3643 0.0156 Bus 9 0.0676 0.2405 0.3056 0.0162 0.3154 0.2014 0.0154 0.0201 0.0141
The algorithms for calculation of sensitivity indices in this thesis are based on modal envelope analysis. Because resonance modes are so close to each other, with several of them not appearing in the modal envelope, the algorithms unfortunately do not work well in this case. It would be possible to overcome this issue by identifying modes for sensitivity index calculation in each modal variable rather than in the envelope. However, this is considered outside the scope of the thesis.
Including the WTG main circuits in the WPP model clearly has a large impact on the modal impedance frequency responses. There are more resonance modes, and the modes are also much more damped. It can be verified that the reduced magnitude of resonance modes is due to resistors in the WTG shunt harmonic filters. In Figure 5.20 there a clus- ter of four resonance modes between 26 pu and 30 pu frequency. Though the sensitivity indices can not be presented, it has been qualitatively observed that these resonance modes are primarily due to array cables and offshore substation transformers resonance. Hence they are similar to the resonance modes which were located between 13 pu and 19 pu before inclusion of the WTG harmonic main circuit, as shown in Figure 5.14. The resonance modes in the 13-14 pu frequency range appear as a result of including the WTG main circuits.
Previously, when the entire wind turbine circuit was modelled as a current source, it did not make a difference whether or not a turbine was considered in operation. Now it is Chapter 5. Offshore WPP Resonance Analysis 72 assumed the grid-side converter and shunt harmonic filter of an out-of-operation WTG is disconnected, leaving the low voltage side of the step-up transformer open-circuited. Hence there is no damping provided by the shunt harmonic filter of an out-of-operation turbine. The modal impedance envelope as a function of turbines in operation obtained
0 2 5
10 1 15 log10|Zm| [pu] 20 0 25 0 5 10 15 20 25 30 35 40 45 50 Frequency [pu] Number of wind turbines
Figure 5.21: Three dimensional plot of modal impedance envelope as fuction of num- ber of turbines in operation. from the 33 bus WPP model is shown in Figure 5.21. The 27 WTG equivalents in this model are disconnected in order of bus number, i.e. the turbine closest to PCCs are disconnected first. The figure illustrates how the modal impedance characteristics of a WPP changes under varying operating conditions. The magnitude of modal impedances are increasingly damped with more turbines in operation, due to the damping effect of the shunt harmonic filters.
The modal impedance scan obtained with the 33 bus WPP model is shown in Figure 5.22. There are some minor differences, but overall the resonance modes are largely similar to those seen in the 9 bus WPP model. Chapter 5. Offshore WPP Resonance Analysis 73
45
40
35
30
25
20
15 Modal impedance Zm [pu] 10
5
0 5 10 15 20 25 30 35 40 45 50 Frequency [pu]
Figure 5.22: Modal impedance scan obtained with 33 bus WPP model, skin effect included and all WTGs connected. The resonance modes are largely similar to the ones obtained from the 9 bus model.
Impact of converter control
With the current controllers tuned as indicated in Section 3.5, the following PI-current control parameters are obtained:
Lmr 0.064 Kp = = 1.5 = 0.169 pu 2Ta,pu 100π · 2500
Lmr 0.064 −1 Ti = τmr = = = 31.8 ms rad = 10 pu (100π) · Rmr 100π · 0.0064 The equivalent impedance representing the main reactor and PI current controller is given by
Kp Zeq(ω) = Rmr + jωLmr + Kp − j ωTi 1 = (Rmr + Kp) + j ωLmr − (3.15 revisited) ω Ti Kp
The modal impedance frequency response with skin effect and all turbines connected is shown in Figure 5.23. Compared to the case in Figure 5.20, where the equivalent impedance of the current controller is neglected, there are some minor changes of modal frequency responses in the range of 5 pu to 20 pu frequency. Chapter 5. Offshore WPP Resonance Analysis 74
9
8
7
6
5
4
3 Modal impedance Zm [pu] 2
1
0 5 10 15 20 25 30 35 40 45 50 Frequency [pu]
Figure 5.23: Modal impedance scan obtained with 9 WPP model, skin effect included, all WTGs connected, and the converter control represented with equivalent impedances. Some small changes can be seen for the modes in the range of 10-15 pu frequency. 6 Discussion
6.1 General Remarks
Frequency responses of the eigenvalues of an admittance matrix allows for locating fre- quencies associated to parallel resonances. Modal switching should be corrected when obtaining the frequency responses. Unlike a traditional impedance or admittance fre- quency scan, a modal impedance scan is not linked to a particular location in the circuit. Rather, the modal resonances are linked to the impedance at defined buses via partici- pation factors.
The weighted participation factor, a term coined by Xu et al. [5], has received little at- tention in published papers. The driving point impedance at some bus for any frequency is the sum of weighted participation factors for the bus in all modal variables. It has been shown in this thesis that when a single WPF approaches the driving point impedance at a resonance frequency, it indicates excitation of a single mode. However, a driving point impedance resonance can also be linked to the excitation of more than one mode. In the WPP analysis, it was shown that it is possible to excite more than one resonance mode simultaneously, i.e. there can be several WPFs of considerable magnitude in a driving point impedance resonance.
In addition to participation factors, a modal impedance sensitivity index and modal fre- quency sensitivity index have been proposed in literature for quantifying the involvement of different network branches in modal resonances. These sensitivity indices indicate the horizontal and vertical movement of the modal impedance peaks, or modes, in the frequency-impedance plane following a small change in some network parameter of in- terest. If the harmonic analysis is carried out with the simplifying assumption of circuit element linearity (i.e. constant values of R, L and C), and all branches of interest are modelled as variation of a series RLC branch, the sensitivity indices can be calculated in a simplified way using eigenvectors obtained during the modal impedance scan. This concept was confirmed for the test cases investigated in Chapter3. It detracts somewhat from the HRMA method that the simplified calculation of the modal sensitivity indices
75 Chapter 6. Discussion 76 breaks down when non-RLC branches and non-linear circuit elements are included in the admittance matrix. The sensitivity indices can still be calculated numerically, but with considerable increase of computational time and effort required.
Some papers have suggested buses with high participation factors are good candidates for placing filters if a particular resonance needs to be mitigated [24, 25]. This is arguably true in the sense that a location with large participation in a resonance mode is where the mode is most observable in the driving point impedance. Beyond this simple fact, the results of this thesis do not imply there is anything extraordinary about buses with high participation in a resonance mode. As has been shown, the resonating branches are not necessarily connected at the bus with highest participation factor, Hence it is in the authors opinion not obvious that a resonance mode can best be mitigated at buses with high participation factor in the mode.
If the frequency of a resonance mode is away from frequencies of harmonic emissions, it may not be of concern. If the resonance mode is close to frequencies of existing harmonic emissions, the frequency sensitivity index can be used to reveal which parameters might be adjusted to move the resonance mode to either higher or lower frequencies. A pa- rameter may however affect several resonance modes. Changing a parameter which has favourable impact on one mode thus might have a negative impact on other modes. Hu et al. [8] combined the frequency sensitivity index with an iterative Newton-Raphson based scheme to adjust the size of selected capacitor banks in small industrial power system, thus placing resonance modes at target frequencies.
There is usually some error tolerance with respect to electrical parameters in production and delivery of power system components such as capacitor banks and filters. Hence it may not be possible to predict the exact location of a resonance mode ahead of time. The frequency sensitivity index might be useful in order to decide what margin there needs to be between the frequency of a predicted resonance mode and frequencies of known harmonic emissions. Alternatively the frequency sensitivity index can be used to determine the error tolerance of electrical components based on the frequency of resonance modes and harmonic emissions.
There is arguably a certain elegance to the modal sensitivity indices. They work well with HRMA because the modes are inherent to the system under study, i.e. not linked to a particular bus. On the contrary, the frequency and magnitude of resonances de- tected via nodal impedance or admittance frequency responses depend on the location at which the scan is obtained. Hence a sensitivity index which is applicable on a system level can not be defined based on traditional methods. However, on a qualitative level Chapter 6. Discussion 77 it is possible to reveal the involvement of different electrical network elements in reso- nances via sensitivity analysis based on traditional impedance and admittance frequency responses [42].
The HRMA described in this thesis is conducted on the positive sequence admittance matrix. Hence it is implicitly assumed the phases are symmetrical, yielding balanced voltages. In reality there is however always some degree of phase unbalance. In a system with considerable unbalance, the resonance frequencies can be different in each phase. Unbalance can occur for a number of reasons. For example, line transpositions can be ineffective for harmonic frequencies. Another cause of phase unbalance might be cables laid in flat formation, i.e. without transpositions. In industry the symmetrical analysis is of limited value due to unbalance being the norm [12]. State of the art software for harmonic analysis such as DigSilent PowerFactory allows for specifying unbalances, hence it can be reflected in frequency scans and in harmonic power flow analysis [15]. If the HRMA method is to find application in industry, it would likely have to be expanded to cover unbalanced systems. Whether the principles of HRMA can be extended to non- symmetrical representation remains an open question.
6.2 Wind Power Plant Analysis
In Chapter5 HRMA was used to analyse a large offshore WPP. The results have not been validated with any real measurements, and so the accuracy of the WPP model is uncertain.
The modes in the studied WPP showed a tendency to group together. In other words several resonance modes are close to each other in terms of resonance frequency and modal impedance magnitude. Resonances in driving point impedance are therefore sometimes linked to excitation of more than one mode, i.e. the driving point impedance can not be estimated by a single WPF. Moreover, in the particular case where there is perfect radial symmetry in the studied system, it was shown that two modal variables can have identical frequency response. Clearly taking only the modal impedance envelope into account is insufficient when studying such large WPPs.
The most critical modes were shown to be associated to resonance between the induc- tances of offshore substation transformers and array collector cable capacitances. The inclusion of frequency dependent resistance (i.e. skin effect) did not cause considerable changes in the frequency of resonance modes, participation factors or sensitivity indices. On the contrary, inclusion of long line corrected cable models had a quite significant impact on frequencies of resonance modes, participation factors and sensitivity indices. Chapter 6. Discussion 78
The frequency and magnitude of resonance modes were shown be highly dependent on the number of turbines in operation. All potential operating conditions must therefore be considered in resonance analysis of such large offshore WPPs.
The analysis showed that the HRMA results are quite sensitive to modelling of the WTG circuit components, i.e. step-up transformers, shunt harmonic filters and main reactors. Hence these components should be included in the resonance analysis. In Section 3.5 it was shown from a theoretical perspective how the current controller of a VSC can influence the impedance as seen from the grid. In the HRMA analysis only a minor effect was observed in the resonance modes when including the current controller.
50
45
40
35
30
25
20
15
10 Resonance mode frequency [pu]
5
0 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 Eigenvalue magnitude [pu]
Figure 6.1: Root locus style presentation of resonance modes observed in 9 bus WPP model with all WTG connected. A small eigenvalue, associated with a large modal impedance, places the mode far to the right in the diagram, indicating poor damping of the resonance.
An alternative presentation of the resonance modes is shown in Figure 6.1. The figure emulates the left half plane of a root-locus diagram, and thus will be familiar to anyone with knowledge of small signal stability analysis. The eigenvalue magnitude and fre- quency of resonance modes are plotted against each other. Resonance modes with little damping have small eigenvalue magnitudes, and are located far to the right in the dia- gram. The frequencies of the resonance modes are indicated on the y-axis. Eigenvalues can not take on negative values, hence modes in the right half plane, which indicate in- stability in small signal analysis, can not occur here. From the root locus style diagram one can clearly see the grouping of the resonance modes. It can also be used to illustrate the movement of the resonance modes following some parameter change in the system. 7 Conclusions
The findings in analysis of test cases were consistent with those presented in published papers. Certain issues in implementation, particularly the issue of modal switching, has not received sufficient attention in past papers, but are critical for meaningful presen- tation and interpretation HRMA results. The combination of resonance modes, partici- pation factors and sensitivity indices can provide a good overview of parallel resonances in a power system.
The finding of this thesis indicate HRMA method can potentially give some guidance for solving problems related to harmonic resonance in offshore WPPs. There is ample potential for resonances to cause problems in offshore WPPs, particularly because the resonance points change with number of turbines in operation. The obtained results were sensitive to modelling of elements within each WTG circuit.
7.1 Further Work
The matrix formulation and methodology suitable for analysis of series resonances has received little attention in literature, and could be investigated further. Also, there is a need to define and verify suitable mitigation strategies based on HRMA. Assump- tions and modelling for HRMA could be verified by comparing results with state of the art simulation software such as DigSilient PowerFactory and/or real measurements. Additionally it would be worth investigating if the HRMA method can be applied to non-symmetrical systems by using a three-phase rather than positive sequence represen- tation in the admittance matrix.
79 A HRMA Test Case Input Data and Results
A.1 Input Data IEEE 14 Bus Test System
Table A.1: IEEE 14 bus test case stedy state load flow [38]
Bus Nominal P Load Q Load LF Voltage LF angle THD # voltage (kV) (pu) (pu) (pu) (deg) (%) 1 230 0 0 1,0600 0,00 1,767 2 230 0 0 1,0450 5,68 2,177 3 230 0 0 1,0427 15,30 1,516 301 35,4 0,50905 0,03363 1,0417 -16,18 9,169 302 35,4 0,50905 0,03363 1,0417 -16,18 9,169 4 230 0,47790 -0,03900 1,0282 -11,41 0,755 5 230 0,07599 0,01599 1,0337 -9,82 1,462 6 230 0 0 1,0700 -15,87 0,468 7 230 0 0 1,0193 -14,47 0,423 8 13,8 0 0,12900 1,0209 -14,49 0,522 9 115 0,29499 16,59900 1,0147 -16,09 0,482 10 115 0,09000 0,05799 1,0168 -16,33 0,421 11 115 0,03501 0,01800 1,0394 -16,21 0,394 12 115 0,06099 0,01599 1,0528 -16,72 0,391 13 115 0,13500 0,05799 1,0458 -16,73 0,376 14 115 0,14901 0,05001 1,0154 -17,39 0,343
Table A.2: IEEE 14 bus test case generator input data [38]
Bus Bus Voltage P gen. Q gen. Sub-transient # Type setting (pu) (pu) (pu) X (pu) 1 Slack 1.0600 2.618681 -0.02863 0.25 2 PV 1.0450 0.18300 5.857 0.25 6 PV 1.0700 -0.11200 44.200 0.25
80 Appendix A. HRMA Test Case Input Data and Results 81
Table A.3: IEEE 14 bus test case branch parameters [38]
Branch From To R1 X1 B1 Type Bus # Bus # (pu) (pu) (pu) Transformer (Y-Y) 4 7 0.00000 0.20900 Transformer (Y-Y) 4 9 0.00000 0.55618 Transformer (Y-Y) 5 6 0.00000 0.25020 Transformer (Y-∆) 7 8 0.00000 0.17615 Transformer (Y-Y) 7 9 0.00000 0.11000 OH Line 6 11 0.09495 0.19887 OH Line 6 12 0.12285 0.25575 OH Line 6 13 0.06613 0.13024 OH Line 9 10 0.03181 0.08448 OH Line 9 14 0.01270 0.27033 OH Line 10 11 0.08203 0.19202 OH Line 12 13 0.22087 0.19985 OH Line 13 14 0.17089 0.34795 OH Line 1 2 0.01937 0.05916 0.05279 OH Line 1 5 0.05402 0.22300 0.04920 OH Line 2 3 0.04697 0.19794 0.04380 OH Line 2 4 0.05810 0.17628 0.03740 OH Line 2 5 0.05693 0.17384 0.03386 OH Line 3 4 0.06700 0.17099 0.03460 OH Line 4 5 0.01335 0.04209 0.01280 Capacitor (pf correction) 9 0 0.00000 0.00000 0.06330 Filter order: 2nd 8 0 0.52510 8.31233 0.03015 Filter order: 5th 8 0 0.52510 1.32635 0.03015 Filter order: 7th 8 0 0.52510 0.67307 0.03015 Filter order: 11th 8 0 0.52510 0.27515 0.03015 Filter order: 11th 3 0 0.00136 0.02772 0.24916 Filter order: 11th 3 0 0.00136 0.02772 0.24916 Transformer (Y-Y) 3 301 0.00000 0.02800 0.00000 Transformer (Y-∆) 3 302 0.00000 0.02800 0.00000 Appendix A. HRMA Test Case Input Data and Results 82
A.2 Test Case Results
Table A.4: Participation factors for all buses and modal resonances in modal scan envelope. The largest participation factor for each frequency is underlined.
Freq [pu] 1.917 2.869 4.212 5.874 8.627 10.738 14.458 25.261 25.943 Zm [pu] 4.4553 9.3417 7.9056 4.5109 6.8356 12.4556 13.1867 22.5070 18.0491 Bus 1 0.0277 0.0556 0.0005 0.0044 0.0427 0.2027 0.0279 0.0027 0.3742 Bus 2 0.0347 0.0742 0.0012 0.0036 0.0364 0.1699 0.0192 0.4026 0.0029 Bus 3 0.0763 0.2632 0.0469 0.0060 0.0013 0.0003 0.0000 0.0078 0.0097 Bus 4 0.0585 0.0972 0.0006 0.0139 0.0455 0.1096 0.0003 0.0976 0.3682 Bus 5 0.0483 0.0802 0.0003 0.0123 0.0542 0.1592 0.0021 0.4138 0.1926 Bus 6 0.0299 0.0257 0.0055 0.0228 0.0464 0.0336 0.0144 0.0306 0.0111 Bus 7 0.1207 0.0769 0.1258 0.1395 0.1074 0.0460 0.0985 0.0025 0.0128 Bus 8 0.2705 0.1000 0.6543 0.4359 0.1149 0.0040 0.0117 0.0005 0.0026 Bus 9 0.0806 0.0555 0.0575 0.0985 0.1376 0.0800 0.3618 0.0051 0.0112 Bus 10 0.0705 0.0467 0.0426 0.0834 0.1202 0.0598 0.2284 0.0008 0.0040 Bus 11 0.0508 0.0351 0.0202 0.0531 0.0870 0.0420 0.0843 0.0048 0.0004 Bus 12 0.0358 0.0258 0.0073 0.0312 0.0577 0.0270 0.0167 0.0174 0.0056 Bus 13 0.0363 0.0264 0.0087 0.0315 0.0561 0.0277 0.0222 0.0137 0.0041 Bus 14 0.0594 0.0377 0.0285 0.0640 0.0927 0.0382 0.1125 0.0001 0.0007 B Details of HRMA
B.1 2-norm of Eigenvectors
Given an n-dimensional vector x1 x2 x = . . xn
The p-norm of a vector is defined as
!(1/p) X p |x|p = |xi| (B.1) i
The 2-norm is equal to one for all eigenvectors returned by the built-in eig()-function in MATLAB. An eigenvector x hence fulfils
2 2 2 |x| = x1 + x2 + ··· + xn = 1 (B.2)
B.2 Modal Sensitivity Indices Differential Expressions
In order to calculate modal frequency sensitivities of a branch in the power network, it 2 2 is necessary to evaluate derivatives ∂Gbr/∂α, ∂ Gbr/∂α∂f, ∂Bbr/∂α and ∂ Bbr/∂α∂f. Many branches, such as first order filters, shunt capacitors, and shunt reactors, can be fully represented by a series RLC branch.
Defining s 2 2 2 1 |Z| = R1 + h X1 − 2 h B1
83 Appendix B. Details of modal analysis 84 and expressing conductance and susceptance of the RLC branch in terms of harmonic order h, resistance R1, inductive reactance X1, and capacitive susceptance B1:
R1 R1 Gbr = 2 = 2 (B.3) 2 2 1 |Z| R + h X1 − 2 1 h B1
1 1 h 2 − X1 h 2 − X1 h B1 h B1 Bbr = 2 = 2 (B.4) 2 2 1 |Z| R + h X1 − 2 1 h B1
Expressions for the first order derivatives of Gbr and Bbr with respect to X1 and B1 are given by 2 1 2h R1 2 − X1 ∂Gbr h B1 = 4 ∂X1 |Z| 2 3 1 2h X1 − 2 ∂Bbr h B1 h = 4 − 2 ∂X1 |Z| |Z|
1 2R1 2 − X1 ∂Gbr h B1 = 2 4 ∂B1 B1 |Z| 2 1 2h X1 − 2 ∂Bbr h B1 1 = 2 4 − 2 2 ∂B1 B1 |Z| hB1 |Z|
The expressions for second order partial derivatives of Gbr and Bbr with respect to har- monic order and either X1 or B1 become highly complex. Expressions for these terms have been obtained using Maple software from Maplesoft, and the expressions are eval- uated in MATLAB. Maple-generated expressions are included for sake of completeness.
2 3 3 3 3 6 3 2 4 2 ∂ Gbr −4Rh B (−B X h + B R Xh + 3BXh − 2) = 2 2 4 2 2 2 2 3 ∂X1∂h (B X h + B R h − 2BXh + 1) 3 3 6 2 2 4 2 2 2 ∂Gbr 4R1hB1(2B1 X1 h − 3B1 X1 h − B1 R1h + 1) = 2 2 4 2 2 2 2 3 ∂B1∂h (B1 X1 h + B1 R1h − 2B1X1h + 1) 2 2 2 4 4 8 4 2 2 6 4 4 4 2 2 4 2 2 2 2 ∂ Bbr −h B (B X h − 6B R X h + B R h − 6B X h + 6B R h + 8BXh − 3) = 2 2 4 2 2 2 2 3 ∂X1∂h (B X h + B R h − 2BXh + 1) 2 4 4 8 4 2 2 6 4 4 4 3 3 6 2 2 4 2 2 2 ∂ Bbr (−3B X h + 6B R X h + B R h + 8B X h − 6B X h − 6B R h + 1) = 2 2 4 2 2 2 2 3 ∂B1∂h (B X h + B R h − 2BXh + 1) C Offshore WPP Analysis
C.1 Anholt WPP Collector Array Layout
LEGEND: Substation WTG Array cables: 150mm Cu cable 240mm Cu cable 500mm Cu cable
Figure C.1
85 Appendix C. Offshore WPP Analysis 86
C.2 Input Data
Table C.1: Data for three-core Cu-conductor submarine array cables of nominal volt- age 33 kV (Um=36 kV) Values taken from ABB user’s guide [40].
Cross section [mm2] 150 240 500 Inductance [mH/km] 0.41 0.38 0.34 Capacitance [µF/km] 0.21 0.24 0.32 Current rating [A] 375 480 655
C.3 Results 9 Bus Model with Nominal Cable Models
Modal resonance at h=12.92 2 ] α
0
-2
-4 Frequency sensitivity [pu/100%] Modal impedance sensitivity [%/100%] Freq sensitivity [pu/% change of -6 Long line 1-2 B Long line 1-2 X Grid reactance. X Long line 2-3 X Long line 6-9 B
Figure C.2: Sensitivty indices of the five parameters with highest frequency sensitivty index with respect to the modal resoance at 12.92 pu frequency. The underground export cable resonates with the grid reactance.
Modal resonance at h=26.597 2 ] α 0
-2
-4
-6
-8
-10 Frequency sensitivity [pu/100%] Modal impedance sensitivity [%/100%] Freq sensitivity [pu/% change of -12 Long line 2-3 B Long line 2-3 X Long line 1-2 B PCC2 transf. 3-5 X PCC1 transf. 3-4 X
Figure C.3: Sensitivty indices of the five parameters with highest frequency sensitivty index with respect to the modal resoance at 26.60 pu frequency. The submarine export cable reactance and shunt capcitance is most involved in this resonace mode. D LCL Filter Design Grid-tied VSC
The design criteria and methodology used for determining filter parameters in this thesis is in accordance with the author’s specialisation project [9] and related paper [10], which again were predominantly based on pre-existing literature [30, 43–45].
Base values given by WTG rated active power and rated AC side voltage are given in table D.1.
Table D.1: Per unit base values wind turbine emission filter design
Parameter Symbol Equation Value Unit Power Sbase = Pn - 3.6 MVA/MW Voltage Vbase - 0.69 kV Frequency f1 - 50 Hz Sbase Current Ibase √ 3012 A 3Vbase 2 Vbase Impedance Zbase 0.13225 Ω Sbase Inductance L Zbase 0.4210 mH base 2πf1 1 Capacitance Cbase 24.07 mF 2πf√1Zbase DC voltage V 2 2√Vbase 1.127 kV dc,base 3
IL1 Ig
L1 C L2
Rd
Figure D.1: LCL filter for grid-tied VSC with nomenclature
With reference to Figure D.1, constraints typically imposed on the LCL filter design include:
87 Appendix D. WTG grid-side LCL filter design 88
1. The converter side inductor L1 should by itself be large enough to limit the ripple current to 10 % of the rated converter current amplitude.
2. The total inductance of L1+L2 should be no larger than 10 % to limit the maximum voltage drop across the filter and subsequent need for high DC side voltage.
3. The reactive power provided by the capacitor should be limited to approximately 5 % of rated power in order to limit reactive power variation seen from the grid.
4. The filter resonance frequency, given by
r 1 L1 + L2 fres = [Hz] (D.1) 2π L1L2C
should be placed in the band of 10f1 ≤ fres ≤ 0.5fsw where fsw is the carrier switching frequency. The is aimed at ensuring the resonance frequency is much lower than that of the lowest carrier band harmonics, and is high enough to not cause controller stability issues.
5. The filter resonance frequency must be damped either passively or actively. Passive damping is the most straightforward solution, but typically results in higher losses and reduced filtering performance compared to active damping. A resistor of magnitude 1 1 Rd = 3 ωresC
where ωres is the filter resonance frequency, will ensure damping of the filter reso- nance.
6. It is generally considered that the LCL filter should attenuate the current ripple √ to 2 % of the rated current amplitude of the converter, i.e. 0.02 2Ibase, in order to comply with relevant grid code.
Some assumptions regarding certain elements of the Anholt offshore wind power plant must be made in order to continue the LCL filter design:
• It is assumed a switching frequency fsw is as high as 2.5 kHz is feasible for the WTG converters.
• It is assumed the converter is able to deliver rated active power of 3.6 MW at a power factor of ±0.9 for maximum active power. This implies a converter rating of 4 MVA.
• An MVA rating of 4.5 MVA with a SC reactance 8 % is assumed for the step up transformer. On the 3.6 MW base of Table D.1 this corresponds to an inductance of 6.4 %. Appendix D. WTG grid-side LCL filter design 89
• It is assumed the semiconductor capacity does not present the binding constraint on the DC side voltage.
The maximum ripple current as function of the converter side inductance L1 can be derived from the converter topology using voltage-second balance. For a two-level con- verter the ripple current as a function of inductor value can be expressed as
Vdc ∆IL1,max = (D.2) 8fswL1 and for a three-level neutral point clamped topology the voltage-second balance across inductor L1 yields Vdc ∆IL1,max = (D.3) 16fswL1 It may be verified that the current ripple at the grid-side of the LCL filter is approxi- mately, 1 ∆Ig,max = ∆IL1,max · (D.4) |1 + L1 (1 − L C(2πf )2) | L2 1 sw
Assuming now that the inductance L2 of the step-up transformer is fixed at 6.4 %, the overall attenuation achieved for different values of inductance L1 and the capacitor C can be plotted as in figure D.2 an figure D.3 for two- and three-level converters respectively. The figures show it is not possible to find a solution adhering to all constraints for either two-level or three-level converter topology. It may also be verified that increasing L2 only makes the situation even worse.
0.08 L1=0.054 pu 0.07 L1=0.064 pu L1=0.074 pu L1=0.084 pu 0.06 L1=0.094 pu
0.05
0.04
0.03 Ripple attenuation [pu] 0.02
0.01
0 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.13 0.14 0.15 Capcitance C [pu]
Figure D.2: Ripple attenuation as function of capacitance C for different sizes of inductor L1, two-level VSC.
The topology of the converter used in Anholt’s turbines is not known, hence an assump- tion must be made. It is therefore assumed that the converter has a 3-point neutral point Appendix D. WTG grid-side LCL filter design 90
0.05 L1=0.054 pu 0.045 L1=0.064 pu L1=0.074 pu 0.04 L1=0.084 pu L1=0.094 pu 0.035
0.03
0.025
0.02
0.015 Ripple attenuation [pu]
0.01
0.005
0 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.13 0.14 0.15 Capcitance C [pu]
Figure D.3: Ripple attenuation as function of capacitance C for different sizes of inductor L1, three-level VSC. clamped topology. Moreover, a trade off must be made to find a filtering solution out- side of the suggested constrained area. Hence it is assumed the grid-side converter can operate with a total inductance somewhat higher than 10 %. Due to the flexible power factor operation of the converter, there is arguably some flexibility on the constraint number three, which limits the size of the capacitor. The desired ripple attenuation of the ripple to 2 % of the peak rated current can thus be achieved with the following sizing of the LCL-filter components,
L1 = L2 = 0.064 pu,C = 11.21 pu, or x = 0.0892 where C = xCbase
The resonance frequency with these parameters is located at the 18.72 pu frequency.
A resistor of magnitude
Rd = 0.220 pu can be installed in series with the shunt capacitor to provide damping of the filter resonance. A resistor in series with the shunt capacitor is a rather crude way to provide damping of the LCL filter resonance, in part due to high losses with such a solution. Better performance can be achieved using active damping or by more sophisticated passive filter design.
The chosen capacitor produces 0.0892 pu or 0.321 MVA reactive power. For high con- verter power output the reactive power contribution from the capacitor is largely offset by filter inductors. The power factor at the grid-side filter terminals can hence be controlled by adjusting the power factor operation of the converter. Appendix D. WTG grid-side LCL filter design 91
More sophisticated filter design methods might reveal that better utilisation of the ca- pacitance can be achieved by specifically targeting the carrier band harmonics which are characteristic for a grid-tied voltage source converter. For example, an inductor might be inserter in series with the capacitor to form an LC branch tuned to the first group of carrier band harmonics, or several LC shunt branches might each be tuned to one group of carrier band harmonics. Such solutions might allow for reducing the size of L1 and C, hence finding a solution within the boundary of the constraints. Such considerations are outside of the scope of this thesis. Bibliography
[1] IEA. Technology roadmap: Wind energy - 2013 edition, 2013.
[2] EY. Offshore wind in europe: Walking the tightrope to success, 2015.
[3] Math H Bollen and Irene Gu. Signal processing of power quality disturbances, volume 30. John Wiley & Sons, 2006.
[4] Lukasz Hubert Kocewiak, Jesper Hjerrild, and Claus Leth Bak. Wind turbine converter control interaction with complex wind farm systems. IET Renewable Power Generation, 7(4):380–389, 2013.
[5] Wilsun Xu, Zhenyu Huang, Yu Cui, and Haizhen Wang. Harmonic resonance mode analysis. Power Delivery, IEEE Transactions on, 20(2):1182–1190, April 2005. ISSN 0885-8977. doi: 10.1109/TPWRD.2004.834856.
[6] Zhenyu Huang, Yu Cui, and Wilsun Xu. Application of modal sensitivity for power system harmonic resonance analysis. Power Systems, IEEE Transactions on, 22 (1):222–231, Feb 2007. ISSN 0885-8950. doi: 10.1109/TPWRS.2006.883678.
[7] Yu Cui and Xiaoyu Wang. Modal frequency sensitivity for power system harmonic resonance analysis. Power Delivery, IEEE Transactions on, 27(2):1010–1017, April 2012. ISSN 0885-8977. doi: 10.1109/TPWRD.2012.2185520.
[8] Haitao Hu, Zhengyou He, Yangfan Zhang, and Shibin Gao. Modal frequency sensitivity analysis and application using complex nodal matrix. Power Deliv- ery, IEEE Transactions on, 29(2):969–971, April 2014. ISSN 0885-8977. doi: 10.1109/TPWRD.2013.2288012.
[9] Henrik Andreas Broch Brantsæter. Specialisation project: Wind Turbine and Off- shore Wind Power Plant Modelling for System Level Harmonic Studies. Norwegian University of Science and Technology (NTNU), 2014.
[10] Henrik Brantsæter,Lukasz Kocewiak, Atle Rygg Ardal,˚ and Elisabetta Tedeschi. Passive filter design and offshore wind turbine modelling for system level har- monic studies. Energy Procedia, 80:401 – 410, 2015. ISSN 1876-6102. doi: 92 Bibliography 93
http://dx.doi.org/10.1016/j.egypro.2015.11.444. 12th Deep Sea Offshore Wind R&D Conference, {EERA} DeepWind’2015.
[11] C. Larose, R. Gagnon, P. Prud’Homme, M. Fecteau, and M. Asmine. Type-iii wind power plant harmonic emissions: Field measurements and aggregation guidelines for adequate representation of harmonics. Sustainable Energy, IEEE Transactions on, 4(3):797–804, July 2013. ISSN 1949-3029. doi: 10.1109/TSTE.2013.2252209.
[12] N.R. Watson J. Arrillaga. Power System Harmonics. John Wiley & Sons, 2 edition, 2003.
[13] Kai Yang. Wind-Turbine Harmonic Emmission and Propagation through A Wind Farm. Master Thesis, Lulea University of Technology, 2012.
[14] Task Force on Harmonics Modeling and Simulation. Modeling and simulation of the propagation of harmonics in electric power networks. Part I: Concepts, models, and simulation techniques. Power Delivery, IEEE Transactions on, 11(1):452–465, Jan 1996. ISSN 0885-8977. doi: 10.1109/61.484130.
[15] DigSILENT PowerFactory 15 User Manual.
[16] K.N.M. Hasan, K. Rauma, P. Rodriguez, J.I. Candela, R.S. Munoz-Aguilar, and A. Luna. An overview of harmonic analysis and resonances of large wind power plant. In IECON 2011 - 37th Annual Conference on IEEE Industrial Electronics Society, pages 2467–2474, Nov 2011. doi: 10.1109/IECON.2011.6119697.
[17] Task Force on Harmonics Modeling and Simulation. Modeling and simulation of the propagation of harmonics in electric power networks. Part II: Sample systems and examples. Power Delivery, IEEE Transactions on, 11(1):466–474, Jan 1996. ISSN 0885-8977. doi: 10.1109/61.484131.
[18] SJ Ranade and W Xu. An overview of harmonics modeling and simulation. IEEE Task Force on Harmonics Modeling and Simulation, page 1, 2007.
[19] Jan Machowski, Janusz W. Bialek, and James R. Bumby. Power System Dynamics Stability and Control. John Wiley & Sons Ltd., 2 edition, 2008.
[20] Yu Cui and Wilsun Xu. Harmonic resonance mode analysis using real symmetrical nodal matrices. Power Delivery, IEEE Transactions on, 22(3):1989–1990, July 2007. ISSN 0885-8977. doi: 10.1109/TPWRD.2007.899481.
[21] Masoud Esmaili, Heidar Ali Shayanfar, and Alireza Jalilian. Modal analysis of power systems to mitigate harmonic resonance considering load models. Energy, 33 (9):1361–1368, 2008. Bibliography 94
[22] K. Rauma, K.N. Md Hasan, C. Gavriluta, and C. Citro. Resonance analysis of a wind power plant with modal approach. In Industrial Electronics (ISIE), 2012 IEEE International Symposium on, pages 2042–2047, May 2012. doi: 10.1109/ISIE. 2012.6237408.
[23] K.N.M. Hasan, K. Rauma, A. Luna, J.I. Candela, and P. Rodriguez. Study on harmonic resonances and damping in wind power plant. In Intelligent and Advanced Systems (ICIAS), 2012 4th International Conference on, volume 1, pages 418–423, June 2012. doi: 10.1109/ICIAS.2012.6306230.
[24] K.N. Binti Md Hasan, K. Rauma, A. Luna, J.I. Candela, and P. Rodriguez. Har- monic compensation analysis in offshore wind power plants using hybrid filters. Industry Applications, IEEE Transactions on, 50(3):2050–2060, May 2014. ISSN 0093-9994. doi: 10.1109/TIA.2013.2286216.
[25] K.N. Md Hasan, A.M. Cantarellas, A. Luna, J.I. Candela, and P. Rodriguez. Grid harmonic detection and system resonances indentification in wave power plant ap- plications. In Industrial Electronics Society, IECON 2013 - 39th Annual Conference of the IEEE, pages 1644–1649, Nov 2013. doi: 10.1109/IECON.2013.6699379.
[26] J. Beerten, S. D’Arco, and J.A. Suul. Cable model order reduction for hvdc systems interoperability analysis. In AC and DC Power Transmission, 11th IET Interna- tional Conference on, pages 1–10, Feb 2015. doi: 10.1049/cp.2015.0039.
[27] Electromagnetic Transient Program Reference Manual (EMTP Theory Book).
[28] J. Wasilewski, W. Wiechowski, and C.L. Bak. Harmonic domain modeling of a distribution system using the digsilent powerfactory software. In Future Power Systems, 2005 International Conference on, pages 7 pp.–7, Nov 2005. doi: 10. 1109/FPS.2005.204272.
[29] Wilsun Xu. Component modeling issues for power quality assessment. Power Engineering Review, IEEE, 21(11):12–17, Nov 2001. ISSN 0272-1724. doi: 10. 1109/MPER.2001.961998.
[30] A.A. Rockhill, M. Liserre, R. Teodorescu, and P. Rodriguez. Grid-filter design for a multimegawatt medium-voltage voltage-source inverter. Industrial Electronics, IEEE Transactions on, 58(4):1205–1217, April 2011. ISSN 0278-0046. doi: 10. 1109/TIE.2010.2087293.
[31] Fritz Santjer, Bernd Weise, Tina Pausch, and Johannes Brombach. Aspects for improvement of measurement and assessment procedures of harmonic emission of wind power plants. In DEWEK 2015, May 2015. Bibliography 95
[32] L. H. Kocewiak, B. L. Øhlenschlæger Kramer, O. Holmstrøm, K. H. Jensen, and L. Shuai. Active filtering application in large offshore wind farms. In Integration of Wind Power into Power Systems as well as Transmission Networks for Offshore Wind Farms, Energynautics GmbH, 11-13 November 2014, Berlin, Germany, pages 1–6, November 2014.
[33] Lukasz Hubert Kocewiak. Harmonics in large offshore wind farms, PhD Thesis. Videnbasen for Aalborg UniversitetVBN, Aalborg UniversitetAalborg University, Det Teknisk-Naturvidenskabelige FakultetThe Faculty of Engineering and Science, 2012.
[34] D.G. Holmes, T.A. Lipo, B.P. McGrath, and W.Y. Kong. Optimized design of sta- tionary frame three phase ac current regulators. Power Electronics, IEEE Transac- tions on, 24(11):2417–2426, Nov 2009. ISSN 0885-8993. doi: 10.1109/TPEL.2009. 2029548.
[35] L. Harnefors, A.G. Yepes, A. Vidal, and J. Doval-Gandoy. Passivity-based con- troller design of grid-connected vscs for prevention of electrical resonance instabil- ity. Industrial Electronics, IEEE Transactions on, 62(2):702–710, Feb 2015. ISSN 0278-0046. doi: 10.1109/TIE.2014.2336632.
[36] F.D. Freijedo, S.K. Chaudhary, R. Teodorescu, J.M. Guerrero, C.L. Bak,L.H. Kocewiak, and C.F. Jensen. Harmonic resonances in wind power plants: Modeling, analysis and active mitigation methods. In PowerTech, 2015 IEEE Eindhoven, pages 1–6, June 2015. doi: 10.1109/PTC.2015.7232382.
[37] Caixia Yang, Kaipei Liu, and Dongxu Wang. Harmonic resonance circuit’s modeling and simulation. In Power and Energy Engineering Conference, 2009. APPEEC 2009. Asia-Pacific, pages 1–5, March 2009. doi: 10.1109/APPEEC.2009.4918398.
[38] R. Burch, G. Chang, M. Grady, E. Gunther, M. Halpin, Y. Liu, M. Marz, T. Ort- meyer, V. Rajagopalan, S. Ranade, P. Ribeiro, T. Sim, and W. Xu. Test systems for harmonics modeling and simulation. Power Delivery, IEEE Transactions on, 14(2):579–587, Apr 1999. ISSN 0885-8977. doi: 10.1109/61.754106.
[39] I Zlatuni´c,S Vodopija, and R Goi´c. Mode switching in modal domain models of overhead lines and underground cables. In International Conference on Power Systems Transients (IPST2015) in Cavtat, Croatia June 15-18, 2015.
[40] ABB. XLPE Submarine Cable Systems, Attachment to XLPE Land Cable Systems - User’s Guide, volume Rev 5. .
[41] ABB. XLPE Land Cable Systems, User’s Guide, volume Rev 5. . Bibliography 96
[42] Vytautas Kersiulis, Andrzej Holdyk, Joachim Holboell, and Ivan Arana. Sensitivity of nodal admittances in an offshore wind power plant to parametric variations in the collection grid. In Proceedings of the 11th International Workshop on Large-Scale Integration of Wind Power into Power Systems, 2012.
[43] A.E. Leon and J.A. Solsona. Performance improvement of full-converter wind tur- bines under distorted conditions. Sustainable Energy, IEEE Transactions on, 4(3): 652–660, July 2013. ISSN 1949-3029. doi: 10.1109/TSTE.2013.2239317.
[44] S.V. Araujo, A. Engler, B. Sahan, and F. Antunes. Lcl filter design for grid- connected npc inverters in offshore wind turbines. In Power Electronics, 2007. ICPE ’07. 7th Internatonal Conference on, pages 1133–1138, Oct 2007. doi: 10. 1109/ICPE.2007.4692556.
[45] A. Reznik, M.G. Simoes, A. Al-Durra, and S.M. Muyeen. Lcl filter design and performance analysis for grid-interconnected systems. Industry Applications, IEEE Transactions on, 50(2):1225–1232, March 2014. ISSN 0093-9994. doi: 10.1109/TIA. 2013.2274612.